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GEOMETRY, DYNAMICS AND SPECTRAL ANALYSIS ON MANIFOLDS The Pestov Identity on Frame Bundles and Eigenvalue Asymptotics on Graph-like Manifolds

EGIDI, MICHELA

How to cite: EGIDI, MICHELA (2015) GEOMETRY, DYNAMICS AND SPECTRAL ANALYSIS ON MANIFOLDS The Pestov Identity on Frame Bundles and Eigenvalue Asymptotics on Graph-like Manifolds, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/11306/

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2 GEOMETRY, DYNAMICS AND SPECTRAL ANALYSIS ON MANIFOLDS The Pestov Identity on Frame Bundles and Eigenvalue Asymptotics on Graph-like Manifolds

Michela Egidi

A Thesis presented for the degree of Doctor of Philosophy

Pure Mathematics Research Group Department of Mathematical Sciences University of Durham England September 2015 To Valerio GEOMETRY, DYNAMICS AND SPECTRAL ANALYSIS ON MANIFOLDS. The Pestov Identity on Frame Bundles and Eigenvalue Asymptotics on Graph-like Manifolds.

Michela Egidi

Submitted for the degree of Doctor of Philosophy September 2015

Abstract

This dissertation is made up of two independent parts. In Part I we consider the Pestov Identity, an identity stated for smooth functions on the tangent bundle of a manifold and linking the Riemannian curvature tensor to the generators of the geodesic ﬂow, and we lift it to the bundle T kM of k-tuples of tangent vectors over a compact manifold M of dimension n. We also derive an integrated version over the bundle P kM of orthonormal k-frames of M as well as a restriction to smooth functions on such a bundle. Finally, we present a dynamical application for the parallel transport of Gk M , the Grassmannian of oriented k-planes of M. In Part II orp q we consider a family of compact and connected n-dimensional manifolds Xε, called graph-like manifold, shrinking to a metric graph as ε 0. We describe the asymp- Ñ totic behaviour of the eigenvalues of the Hodge Laplacian acting on diﬀerential forms on Xε in the appropriate limit. As an application, we produce manifolds and families of manifolds with arbitrarily large spectral gaps in the spectrum of the Hodge Laplacian. Declaration

The work in this thesis is based on research carried out at the Pure Mathematics Research Group, the Department of Mathematical Sciences, England. No part of this thesis has been submitted elsewhere for any other degree or qualiﬁcation. Part I and Appendix A are all my own work unless referenced to the contrary in the text. Part II is based on joint work with Olaf Post (Trier University). In particular, Chapter 7 is my own independent contribution unless referenced otherwise, and Chapter 8 contains our joint work.

Copyright c 2015 by Michela Egidi. “The copyright of this thesis rests with the author. No quotations from it should be published without the author’s prior written consent and information derived from it should be acknowledged”.

iv Acknowledgements

I am indebited with my two supervisors Norbert Peyerimhoﬀ and Olaf Post for their kind attention and the innumerable and endless discussions. I also would like to thank Gerhard Knieper for giving me access to an unpublished result [Kni] of his, Vicente Cort´es for providing the references in Remark 4.2.7, and Bruno Colbois for pointing Olaf’s and my attention to the McGowan Lemma in [McG93].

v Contents

Abstract iii

Declaration iv

Acknowledgements v

I 1

1 Introduction 2 1.1 ThePestovIdentityanditsapplications ...... 2 1.2 Aimandmainresults...... 9 1.3 Overviewofthetext ...... 11

2 A new framework 13 2.1 The bundles T kM and P kM ...... 13 2.1.1 The new bundle T kM ...... 13 2.1.2 The frame bundle P kM ...... 14 2.2 FrameFlows...... 16 2.3 DiﬀerentialOperators...... 17

3 ThePestovIdentityonFrameBundles 20 3.1 TheLiftedPestovIdentity ...... 20 3.2 TheIntegratedPestovIdentity ...... 25 3.2.1 Restriction to functions on the frame bundle P kM ...... 28

vi Contents vii

4 Dynamical Applications 33 4.1 An invariance property of smooth functions on frame bundles .... 33 4.2 An invariance property of smooth functions on oriented Grassmannians 34 4.2.1 Oriented Grassmannians and Parallel Transport ...... 35 4.2.2 InvarianceProperty...... 37

II 41

5 Introduction 42 5.1 Aimandmainresults...... 43 5.2 Overviewofthetext ...... 46

6 Preliminaries 47 6.1 GraphsandtheirLaplacians ...... 47 6.2 MetricGraphsandtheirLaplacians ...... 49 6.3 RamanujanGraphs ...... 52 6.4 Graph-like Manifolds and their Hodge Laplacian ...... 54 6.5 HodgeTheory...... 57 6.6 Someusefulfactsabouteigenvalues ...... 59

7 Asymptotic behaviour 63 7.1 Harmonicforms...... 64 7.2 Convergence for functions and exact 1-forms ...... 65 7.3 Divergence for co-exact p-forms ...... 67 7.3.1 Eigenvalue asymptotics on the building blocks ...... 68 7.3.2 Maintheorems ...... 70

8 Manifolds with spectral gap 74 8.1 Hausdorﬀ convergence of the spectrum and spectral gaps ...... 74 8.2 Manifolds with arbitrarily large spectral gap ...... 76 8.3 Families of manifolds with special spectral properties arising from fam- iliesofgraphs ...... 79 Contents viii

Appendix 87

A Geometry of T kM 87 A.1 Horizontalandverticallifts ...... 88 A.2 LieBrackets...... 90 A.3 CovariantDerivative ...... 92 A.4 TheRiemanniancurvaturetensor ...... 97 A.5 Curvature ...... 100

Bibliography 104 List of Figures

4.1 From left to right: example of intrinsic and non-intrinsic parallel trans-

port of a plane Aor along the geodesic cv...... 36

6.1 An example of a two dimensional graph-like manifold Xε with associ-

ated graph X0...... 55

8.1 (a) Region where the 0-form eigenvalue convergence in (8.2.3b) holds

0 i 2β i (max β, 0 α 2); (b) Region where λ X h ν λj X (β t u ă { j p εi q i p 0q ą α 2, β 0 or β α 4, β 0); (c) Region where λ¯p Xi diverges. ´ { ě ą´ { ď 1p εi q (α 1 2,α β or 4α 2β 1 0, α β); (d) Blue region: all ą { ě ´ ´ ą ď eigenvalues diverge. Green region: the form eigenvalues diverge and the function eigenvalues converge to 0. Above the red dotted line the volume tends to 0, below it tends to ...... 81 8 8.2 Parameter region where the rescaled 0-form eigenvalue λ0 Xi di- 1p εi q verges, i.e., region where max β, 0 α 2 and λ0 Xi h ν2βλ Xi t u ă { 1p εi q r 1p 0q are both satisﬁed. Above the dotted lines, the exponent in (8.3.6) is r positiveandtheeigenvaluediverges...... 83 8.3 Above the dotted lines the p-forms eigenvalues diverge (2 p n 1), ď ď ´ dependingonthedimension...... 85

ix Part I

1 Chapter 1

Introduction

The Pestov Identity is an identity stated for smooth functions on the tangent bundle of a Riemannian manifold M. It links the generator of the geodesic flow with the Riemannian curvature tensor and other geometrically motivated differential operators and, therefore, it can be considered as a dynamical Weitzenböck identity on the tangent bundle. It was first introduced by Pestov and Sharafudtinov in [PS88] to derive useful estimates on symmetric tensor fields and to give an answer to the question whether a smooth symmetric tensor field can be uniquely recovered from the knowledge of all its integrals along geodesics. Since then, it has been widely used to solve Geometric Inverse Problems such as tensor tomography, the boundary rigidity problem and spectral rigidity. In this dissertation we lift this identity to the space of k-tuples of tangent vectors over a compact n-dimensional manifold M and we restrict it to the principal bundle of orthonormal k-frames. As an application, we use it to obtain an invariance property of smooth functions on Grassmannians under the parallel transport.

1.1 The Pestov Identity and its applications

Let M,g be a compact manifold of dimension n and let TM and SM be its tangent p q bundle and unit tangent bundle, respectively. Let π : TM M be the canonical ÝÑ projection of TM onto M. TM is a 2n-dimensional manifold whose tangent space

2 1.1.ThePestovIdentityanditsapplications 3 at v TM splits as P TvTM Hv Vv TpM TpM, “ ‘ – ˆ where Hv and Vv are called horizontal and vertical distributions at the point v, respectively. In fact, let X : ε,ε TM be a curve in TM with X 0 v, and p´ q ÝÑ p q“ let π X be its footpoint curve on M. Then, ˝

1 d D TvTM X 0 π X t , X t , Q p q– dt t“0p ˝ qp q dt t“0 p q ´ ˇ ˇ ¯ D ˇ ˇ where is the covariant derivative alongˇ π X. ˇ dt ˝ Hence, we deﬁne the two distributions as follows.

1 D Hv X 0 TvTM X t 0 w, 0 w TπpvqM TπpvqM, “ p q P dt t“0 p q“ – tp q | P u– ˇ ˇ ( 1 dˇ ˇ Vv X 0 TvTM ˇ π X t 0 0,w w TπpvqM TπpvqM. “ p q P dt t“0p ˝ qp q“ – tp q | P u– ˇ ˇ ( h v h Therefore, every vectorˇ ξˇ TvTM splits uniquely as ξ ξ ξ with ξ Hv and ˇP “ ` P v ξ Vv, called horizontal and vertical component, respectively. P We equip TM with the Sasaki metric [Dom62,GuKa02], deﬁned as

h h v v ξ,η TTM ξ ,η TM ξ ,η TM . (1.1.1) x y “ x y ` x y The structure of TTM gives rise to horizontal and vertical diﬀerential operators, deﬁned below.

8 Let ψ C TM and denote by uw t the parallel transport of the vector u along P p q p q the geodesic cw : ε,ε M with starting point cw 0 π w and starting vector p´ q ÝÑ p q“ p q h v c1 0 w. The gradient of ψ at v TM is given by gradψ v gradψ v , gradψ v wp q“ P p q “ p p q p qq where horizontal and vertical component are deﬁne intrinsically as h d v d gradψ v ,w ψ vw t and gradψ v ,w ψ v tw . x p q y“ dt t“0 p p qq x p q y“ dt t“0 p ` q ˇ ˇ ˇ ˇ In other words, theyˇ describe the derivative of ψ along the horizontalˇ curve t ÞÑ vw t and along the vertical curve t v tw. p q ÞÑ ` Let X : TM TM be a semi-basic vector ﬁeld, i.e., an element of X π˚ TM , ÝÑ p p qq π˚ TM being the pullback bundle of the vector bundle TM over M via the projection p q π : TM M. The horizontal and vertical covariant derivative of X are given by ÝÑ h D v D ∇wX v X vw t and ∇wX v X v tw . p q“ dt t“0 p p qq p q“ dt t“0 p ` q ˇ ˇ ˇ ˇ ˇ ˇ 1.1.ThePestovIdentityanditsapplications 4

Consequently, the horizontal and vertical divergence of X are

h n h v n v

divX v ∇ei X v ,ei and divX v ∇ei X v ,ei , p q“ i“1x p q y p q“ i“1x p q y ÿ ÿ where ei,...,en is an orthonormal basis of TpM for p π v . “ p q Finally, we define the geodesic flow on TM. Let cv : ε,ε M be the geodesic p´ q ÝÑ with initial vector c1 0 v. The geodesic flow is the map vp q“ φt : TM TM, such that φt v c1 t , (1.1.2) ÝÑ p q“ vp q and its generator is the vector

d t d t XG v φ v π φ v , 0 v, 0 , p q“ dt t“0 p q– dt t“0p ˝ qp q “ p q ˇ ´ ˇ ¯ i.e., XG v is a horizontal vector.ˇ ˇ p q ˇ ˇ Theorem 1.1.1 (The Pestov Identity, [Kni02]). Let M be a compact n-dimensional Riemannian manifold. For all ψ C8 TM , we have P p q h v h h v 2 2 gradψ v , grad XGψ v gradψ v divY v divZ v x p q p p qqy “ } p q} ` p q` p q v v (1.1.3) R gradψ v ,v v, gradψ v , ´ x p p q q p qy where h v h v Y v gradψ v , gradψ v v v, gradψ v gradψ v , p q “ x p q p qy ´ x p qy p q and h h h Z v XGψ v gradψ v v, gradψ v gradψ v . p q“ p q ¨ p q “ x p qy p q The reader will ﬁnd a coordinate-free proof in [Kni02, Appendix] and the original coordinate-based proof in [PS88] or [Sha94], both presented for manifolds of any dimension. Most of the applications use the integrated version of this identity, where integration is performed over SM with respect to the Liouville measure dµL.

Theorem 1.1.2 (Integrated Pestov’s Identity, [Kni02]). Let M be a compact n- dimensional Riemannian manifold and ψ C8 TM . Then, P p q h v 2 gradψ v , gradXGψ v dµL x p q p qy “ żSM h 2 2 gradψ v dµL n 1 XGψ v dµL (1.1.4) SM } p q} ` p ´ q SM p p qq ż v ż v v ∇vZ v ,v dµL R gradψ v ,v v, gradψ v dµL, ` x p q y ´ x p p q q p qy żSM żSM 1.1.ThePestovIdentityanditsapplications 5 where Z v is as in Theorem 1.1.1. p q The proof can be found in [Kni02, Theorem 1.1, Appendix]. It is based on the fact that the horizontal divergence vanishes under integration over SM while integration of the vertical divergence produces the second and third terms in the RHS (see [Kni02, Lemma 1.2, Appendix]). The dynamical component of (1.1.3) and (1.1.4) lies in the presence of the generator of the geodesic flow, which is linked to the sectional curvature of the plane v span gradψ v ,v . This indicates a close connection between properties of the geodesic t p q u flow and the curvature of the manifold. Such a connection is already well known in relation to ergodicity of the geodesic flow, see for example [Bal95]. Moreover, since the norm of the horizontal gradient (the first term in the RHS of (1.1.3)) is related to parallel transport, Theorems 1.1.1 and 1.1.2 provide a link between curvature conditions and invariant properties of functions under the action of the geodesic flow and under parallel transport. Despite the strong dynamical flavour of these identities, their main applications are not in the field of pure Dynamics but in Integral Geometry and Inverse Problems. Below, we present a brief overview of three problems where the Pestov Identity plays a key role for the solution.

Tensor tomography. The material here presented has been extracted from [PSU13, PSU14a]. Tensor tomography is a subﬁeld of integral geometry that studies how to recover a function or a tensor ﬁeld by the knowledge of its integrals along curves. The simplest example is the X-ray (or Radon) transform in the plane, which aims at recovering a function f in R2 studying the integral of f along straight lines, i.e., geodesics in R2. This classical problem is nowadays well-known and well-studied. We refer the reader to [Hel11] for its properties and further information. In the context of Riemannian manifolds, the problem of tensor tomography, or the geodesic ray transform problem, is posed as follows. Let M,g be a compact, oriented Riemannian manifold of dimension n 2 with p q ě boundary, and let ν be the unit outer normal to the boundary M of M. Let SM B 1.1.ThePestovIdentityanditsapplications 6

be its unit tangent bundle and let p,v be a point in SM, i.e., v SpM, then p q P SM SM SM , Bp q “ B`p q B´p q ď where SM p,v SM ν p ,v 0 . B˘p q“tp q P Bp q | ¯ x p q yě u Without loss of generality, we think of M as embedded into a compact n-dimensional manifold N without boundary. The exit time τ : SM 0, of a unit speed N- ÝÑ r 8s geodesic γt p,v is τ p,v inf t 0 γt p,v N M . If the geodesic γt p,v never p q p q“ t ą | p q P z u p q leaves the manifold M, we deﬁne τ p,v . In the case γt p,v , the manifold p q “ 8 p q ă 8 M is called non-trapping and the geodesic ray transform of a function f C8 SM P p q is then deﬁned as

τpp,vq If p,v f φt p,v dt, p,v SM , p q“ p p qq p q P B`p q ż0 where φt is the geodesic ﬂow on M, and the geodesic ray transform on a symmetric m-tensor F is deﬁned as ImF : Ifm, where fm is the function on SM arising from “ the tensor F via fm p,v F p,v ,..., p,v . p q“ pp q p qq Given a smooth function f or a smooth m-tensor F , the geodesic ray transform problem explores what properties of f or F can be recovered from the knowledge of

If or ImF . It is known [Sha94] that a sufficiently smooth tensor field F can be composed into a solenoidal and potential part, denoted by f s and dp, respectively, i.e., F f s dp, “ ` where f s is a divergence free m-tensor field and p is a smooth m 1 -tensor field p ´ q vanishing at the boundary. Using integration by parts and the fact that p vanishes on the boundary, it is easy to see that the geodesic ray transform of dp, the potential part of f, vanishes. Therefore, we can only aim at recovering the solenoidal part of f, which justifies the notion of s-injectivity, defined as follows. The X-ray transform on symmetric m-

s tensor ﬁelds, m 1, is s-injective if ImF 0 implies f 0 for any smooth m-tensor ě “ “ F . If m 0, i.e., in the case of functions, I is s-injective if I f 0 implies f 0 “ 0 0 “ “ for any f C8 SM . P p q A number of results are known about s-injectivity. I0 and I1 are s-injective

[Muk77, AR97]. Im is s-injective for all m on simple surfaces [PSU13], i.e., surfaces with strictly convex boundary and such that for any two points there exists a unique 1.1.ThePestovIdentityanditsapplications 7

geodesic joining them and depending smoothly on the end-points. Moreover, Im is known to be s-injective on manifolds of negative curvature [PS88], under other curvature restrictions [Sha94], or on higher dimensional simple or Anosov manifolds with certain conditions on modified Jacobi fields [PSU15]. I2 is s-injective for manifolds of any dimension equipped with simple metrics including real-analytic ones [SU05b]. We remind the reader that a manifold M is said Anosov if the linearisation Dφt of its geodesic flow splits the tangent bundle of SM into tree invariant subspaces: a one- dimensional subspace tangent to the direction of the flow, and two other subspaces on which Dφt acts uniformly contracting and expanding, respectively. In addition, tensor tomography has also been studied in other contexts. For example, it has been considered in the presence of an attenuation factor [SU11,PSU12], in the presence of a magnetic field (see [DP05,Ain13] and references therein), and for thermostats [DP07]. The main idea of the injectivity proof lies in observing that the transport equation XGu f in SM with u SM 0 is solved by the function u x,v “ ´ |Bp q “ p q “ τpx,vq f φt x,v dt. Therefore, it is enough to prove that u is constant, as this al- 0 p p qq şready implies f 0 by the boundary condition. Then, the Pestov Identity enters “ 2 the game giving an estimate of XGu that, together with other tools, allows us to } } conclude that u 0. “

The boundary rigidity problem. The material presented in this paragraph has been extracted from [SU05a]. The boundary rigidity problem addresses the question whether it is possible to recover uniquely the metric of a Riemannian manifold from the knowledge of the geodesic distance between any two points on the boundary. This problem arises in geophysics in an attempt to determine the inner structure of the Earth by measuring the travel times of seismic waves. We observe that one can construct a metric g on a manifold M with boundary and

ﬁnd a point x M such that dg x , M sup dg x,y . For such a metric, 0 P p 0 B q ą x,yPBM p q dg is independent of a change of g in a small enough neighbourhood of x0 [Uhl]. Therefore, it is natural to pose restrictions on the metric we want to recover. In 1981 1.1.ThePestovIdentityanditsapplications 8

Michel [Mic81] conjectured that every simple manifold is boundary rigid. Pestov and Uhlmann [PU05] proved the conjecture for simple two-dimensional manifolds with no restriction on the curvature. For higher dimensional manifolds, results are known for ﬂat metrics, conformal metrics, and for locally symmetric spaces with negative curvature. We refer the reader to the survey [SU05a] and references therein. There is a strong link between the boundary rigidity problem and the operator

Im introduced in the previous paragraph. In fact, the linearisation of the boundary rigidity problem near a simple metric g is given by showing that I2 is s-injective on symmetric 2-tensor ﬁelds [Sha94]. Therefore, the Pestov Identity is a tool often used in proofs in a fashion similar to the one described in the previous paragraph (see for example [SU00] and [Dar06]). As for tensor tomography, the boundary rigidity problem has been considered for other types of dynamics such as the magnetic ﬂow. For results in this direction, we refer the reader to [DPSU07] and references therein.

Spectral rigidity. The material presented here has been extracted from [CS98].

Let M,g be a closed Riemannian manifold without boundary and let gt t ε,ε p q t u Pr´ s be a family of metrics with g g and smoothly depending on t such that the spectra 0 “ of the Laplacian on M,gt coincide. Such a family is called isospectral deformation. p q Spectral rigidity is concerned with the question whether every isospectral deformation comes from a family of diffeomorphisms ϕt : M M such that ϕ0 id and ÝÑ “ t ˚ gt ϕ g . The manifold M is spectrally rigid if for all isospectral deformations “ p q 0 t gt t there exists a family of diffeomorphisms ϕ : M M smoothly depending on t u ÝÑ 0 t ˚ t and such that ϕ id and gt ϕ g . “ “ p q 0 This problem was initially posed by Guillemin and Kazhdan in [GK80a] where they proved that a two-dimensional, closed and negatively curved manifold is spectrally rigid. In a subsequent paper, they extended this result to manifolds of any dimension under a curvature pinching assumption [GK80b]. In both papers, the claim follows from the s-injectivity of I0. Min-Oo [Min86] proved it for manifolds with negative definite curvature operator. These three results were proved without the use of the Pestov Identity. However, it appeared as a key feature in [CS98] where Croke 1.2. Aim and main results 9 and Sharafutdinov used it to prove that any closed negatively curved manifold of dimension n is spectrally rigid. Their result is again a consequence of the s-injectivity of the operator Im on compact manifold of negative curvature as in [GK80a,GK80b], but they give an alternative proof of s-injectivity using the Pestov Identity. Spectral rigidity has also been considered for a wider class of manifolds, namely, Anosov manifolds. Sharafutdinov and Uhlmann [SU00] proved that an Anosov surface with no focal points is spectrally rigid. This result relies on the s-injectivity of

I2 for Anosov surfaces with no focal points. More recently, Paternain, Salo and Uhlmann [PSU14b] extended this result to all closed oriented Anosov surfaces. Again, this is a consequence of the fact that I2 is s-injective on a closed oriented Anosov surface.

1.2 Aim and main results

The aim of our investigation is to derive a Pestov-type identity for smooth functions on the bundle T kM of k-tuples of tangent vectors over a compact n-dimensional manifold M and restrict it to the principal bundle P kM of orthonormal k-frames, which we think of as a subspace of T kM. More precisely, the two bundles are deﬁned as (see also Section 2.1)

k T M : TpM ... TpM, “ pPM ˆ ˆ ď k´times k k k P M v ,...,vk TloooooooooomoooooooooonM vi,vj δij T M. “ tp 1 q P x y“ uĂ ˇ ˇk k In particular, it is possible to describe Tf T M, f T M, mocking the splitting of P TTM into horizontal and vertical component. This also apply to the tangent space of P kM, where the splitting appears naturally (see Section 2.1 or [KN63] for a general overview on principal bundles). This allows us to define horizontal and vertical differential operators following the description of Section 1.1 (see Section 2.3). Regarding the dynamics we use, it is given by the frame flows, the lifts of the geodesic flow, defined in the following way.

k Let f v ,...,vk T M and choose the vector vi for some i. The i-th frame ﬂow, “ p 1 q P 1.2. Aim and main results 10

i 1,...,k, is the map that parallel transports f along the geodesic cv with starting “ i vector vi, i.e., every vector vj component of f is parallel transported along cvi . Such a framework allows us to state the Lifted Pestov Identity (Theorem 3.1.4, Section 3.1), and its integrated version (Theorem 3.2.3, Section 3.2) for smooth functions on T kM. For sake of brevity we do not restate them here. However, the real new results are the restriction of Theorem 3.2.3 to smooth functions on the principal bundle P kM (Theorem 3.2.4, Subsection 3.2.1) and an equality for smooth functions on P kM being invariant under one of the frame flows (Corollary 3.2.6, Subsection 3.2.1), where only the L2-norm of the generators of the frame flows and the Riemannian curvature tensor are involved. In particular, Corollary 3.2.6 is the key identity for our applications. In fact, it appears that if the manifold M is negatively curved, then any function invariant under one of the frame flows might also be invariant under the remaining ones. This is true when M is a two-dimensional negatively curved manifold or M is a n-dimensional manifold with constant sectional curvature (see Section 4.1). However, the main application is for smooth functions on oriented k-th Grass- mannians Gk M , k 1,...,n, i.e., Grassmannians where the k-planes come with orp q “ an intrinsic orientation (for a precise definition we refer the reader to Section 4.2). The bundle P nM projects canonically onto Gk M and every function on Gk M orp q orp q can be lifted to a function on P nM. Moreover, on oriented Grassmannians we distinguish between intrinsic and non- intrinsic parallel transport of oriented k-planes. The parallel transport of an oriented k-plane Aor is called intrinsic if it is along a geodesic cv with starting vector v Aor, P it is called non-intrinsic, otherwise (see also Definition 4.2.2 in Section 4.2). Due to the projection of P nM onto Gk M , every smooth function on Gk M invariant orp q orp q under the parallel transports has a smooth lift on P nM which is invariant under the first k frame flows. This link allows us to prove the following theorem via Corollary 3.2.6.

Theorem 1.2.1. Let M be a compact n-dimensional Riemannian manifold with non- positive curvature operator (R 0). Let 1 k n and ϕ C8 Gk M . If ϕ is ď ď ď P p orp qq invariant under all the intrinsic parallel transports then it is also invariant under all 1.3. Overview of the text 11 parallel transports.

For the deﬁnition of the curvature operator, we refer the reader to (4.2.2) in Section 4.2. We also point out that for k 1, G1 M SM, the intrinsic parallel transport “ orp q “ corresponds to the geodesic ﬂow and the non-positivity of the curvature operator of M relaxes to the non-positivity of the curvature of M. Therefore, our result yields the following, which recovers an unpublished result of Knieper [Kni].

Corollary 1.2.2. Let M be a compact Riemannian manifold with non-positive curvature. Let ϕ C8 SM invariant under the geodesic ﬂow, then ϕ is also invariant P p q under parallel transport.

Finally, combining the above theorem with Berger’s holonomy classiﬁcation, we obtain the following proposition.

Proposition 1.2.3. Let M be a non-ﬂat, compact Riemannian manifold with non- positive curvature operator R. Then, the following statements hold:

(i) If M is either a K¨ahler or a Quaternion-K¨ahler manifold of real dimension 2n 4 or 4n 8, respectively, or a locally symmetric space of non-constant ě ě curvature (i.e., not the real hyperbolic space), then there exist smooth, non- constant functions on G2 M or G4 M which are invariant under all intrinsic orp q orp q parallel transports.

(ii) If M is not one of the exceptions in i , then, for all k dim M, any smooth p q ď function on Gk M which is invariant under intrinsic parallel transport is nec- orp q essarily constant.

1.3 Overview of the text

The material is organized as follows. In Chapter 2 we describe the bundles T kM and P kM together with their diﬀerential operator and the frame ﬂows. In Chapter 3 we present our main results. In Sections 3.1 and 3.2 we state and prove the Lifted Pestov Identity and the Integrated Lifted Pestov Identity, respectively. Moreover, in 1.3. Overview of the text 12

Subsection 3.2.1, we restrict the Integrated Lifted Pestov Identity to smooth functions on P kM. In Chapter 4 we present dynamical applications on smooth functions on P nM over a two-dimensional manifold of negative sectional curvature and over a n- dimensional manifold of constant sectional curvature invariant under one of the frame ﬂows, and on smooth functions on oriented Grassmannian invariant under intrinsic parallel transports. Chapter 2

A new framework

We here introduce the two spaces we will work with in the next chapter, together with some of their features, the description of their differential operators and the dynamics we equip it with. The new spaces are the bundles T kM and P kM already introduced in Section 1.2. This chapter is structured as follows. In Section 2.1 we describe the bundles T kM and P kM, their tangent spaces and the chosen metric on them. In Section 2.2 we explain how the geodesic flow and the frame flows are related and we describe the generators of the latter. Finally, in Section 2.3 we describe the geometrically motivated differential operators related to the structure of the tangent spaces of T kM and P kM, namely, horizontal and vertical gradient, covariant derivative and divergence.

2.1 The bundles T kM and P kM

2.1.1 The new bundle T kM

Let M,g be a compact Riemannian manifold of dimension n. Let TM be its p q tangent bundle and π : TM M, v p if v TpM, be the canonical projection. ÝÑ ÞÑ P Let 1 k n, we deﬁne the space of k-tuples of tangent vectors over M as ď ď

k T M : TpM ... TpM . “ pPM ˆ ˆ ď k´times loooooooooomoooooooooon 13 2.1. The bundles T kM and P kM 14

k k This space projects canonically onto M via π : T M M, f u ,...,uk ÝÑ “ p 1 q ÞÑ p if p π ui for all i 1,...k. “ p q “ T kM is a manifold of dimension kn n (for more details on its geometry, see ` Appendix A).

k k Let f v ,...,vk T M and let X V ,...Vk : ε,ε T M be a “ p 1 q P “ p 1 q p´ q ÝÑ k curve on T M such that X 0 f and that the Vi’s are all vector ﬁelds on M along p q“ the footpoint curve πk X on M. Then, ˝

k 1 d k D D Tf T M X 0 π X t ; V1 t ,..., Vk t . Q p q“ dt t“0p ˝ qp q dt t“0 p q dt t“0 p q ´ ˇ ˇ ˇ ¯ ˇ k ˇ ˇ Therefore, the tangent spaceˇ of T M at f is givenˇ by ˇ

k Tf T M T k M ... T k M . (2.1.1) “ π pfq ˆ ˆ π pfq pk`1q´times

loooooooooooooomoooooooooooooonk We call the ﬁrst TπkpfqM copy in Tf T M horizontal distribution and the product of the remaining k copies of TπkpfqM vertical distribution. Consequently, any vector k h v x Tf T M is written as the sum of x x ;0,..., 0 and x 0; x ,...,xk , called P “ p 0 q “ p 1 q horizontal and vertical component, respectively. This construction allows us to deﬁne a Sasaki-type metric on T kM.

k Let x x ; x ,...,xk ,y y ; y ,...,yk Tf T M. Then, “ p 0 1 q “ p 0 1 q P k

x,y k : x ,y x ,y . (2.1.2) Tf T M 0 0 TπkpfqM i i TπkpfqM x y “ x y ` i“1x y ÿ Consequently, the horizontal and vertical distribution are pairwise orthogonal.

2.1.2 The frame bundle P kM

Let M,g be as in Subsection 2.1.1. The frame bundle of orthonormal k-frames over p q M is denoted by

k k k P M v ,...,vk T M vi,vj δij T M. “ tp 1 q P x y“ uĂ ˇ The orthogonal group O k acts on the rightˇ on this space. p q As for T kM, P kM projects canonically onto M and the projection map is again

k denoted by π . This map is a ﬁbration where the ﬁbre Fp is the Stiefel manifold of n orthonormal k-frames over R , i.e., Fp O n O n k . – p q{ p ´ q 2.1. The bundles T kM and P kM 15

In particular, for k 1 we have P 1M SM, where SM denotes the unit tangent “ “ bundle of M. On the other hand, when k n, P nM is a principal bundle with fibre “ isomorphic to O n . p q k k Let f v ,...,vk P M. Any curve in P M through the point f is given “ p 1 q P k by X V ,...,Vk : ε,ε P M where the Vi’s are orthonormal vector fields “ p 1 q p´ q ÝÑ k 1 k along the footpoint curve π X with Vi 0 vi. X 0 Tf P M is described as in ˝ p q“ p q P D (2.1.1) with an additional condition on the Vi t ’s. Since Vi t ,Vj t δij for dt t“0 p q x p q p qy “ all t, differentiation at t 0 yields ˇ “ ˇ D D Vi t ,vj Vj t ,vi . xdt t“0 p q y “ ´xdt t“0 p q y ˇ ˇ ˇ ˇ Therefore, the tangent spaceˇ of P kM at f isˇ given by

k Tf P M u; w ,...,wk TpM ... TpM wi,vj o k , (2.1.3) “ p 1 q P ˆ ˆ x y ij P p q ! ˇ ) ˇ ` ˘ with o k the Lie algebra of O k , i.e, the set of skew-symmetricˇ real matrices of p q p q k dimension k k. Tf P M splits orthogonally into a horizontal and a vertical distri- ˆ P P bution, Hf and Vf , described below.

P H u;0,..., 0 TpM ... TpM TpM, f “ p q P ˆ ˆ – ! ) P V 0; w ,...,wk TpM ... TpM wi,vj o k . f “ p 1 q P ˆ ˆ x y ij P p q ! ˇ ) We observe that for k n, the vertical distributionˇ ` is isomorphic˘ to o n . “ ˇ p q k h v h P Consequently, any vector u Tf P M splits as u u u where u H and P “ ` P f v P u 0; u ,...,uk V called again horizontal and vertical component, respectively. “ p 1 q P f We point out that vectors on T kM are not automatically vectors on P kM, as they do not satisfy the constrain in (2.1.3). To obtain a vector on P kM from a vector on T kM, we need to perform an orthogonal projection of the vertical components

k of the latter onto Tf P M, or more precisely, onto the Lie algebra o k . We give an p q example below.

k k Let f v ,...,vk P M and X f ; Y f ,...,Yk f Tf T M. Then, the “ p 1 q P p p q 1p q p qq P k vector X f ; Y ,o f ,...,Yk,o f Tf P M is deﬁned component-wise as follows. p p q 1 p q p qq P 1 k Y o f : Y f Y f ,v Y f ,v v . (2.1.4) i, i 2 i j j i j p q “ p q´ j“1 x p q y ` x p q y ÿ ´ ¯ 2.2. Frame Flows 16

It is easy to check that the matrix Yi,o f ,vj is skew-symmetric. x p q y ij Finally since P kM is a submanifold` of T kM, it˘ inherits the metric described in (2.1.2) and horizontal and vertical component are again pairwise orthogonal.

2.2 Frame Flows

We now introduce the frame flows F i, i 1,...,k. t “ 1 k The first frame flow Ft on T M is the lift of the geodesic flow and, more generally, i k the i-th frame flow F on T M is the parallel transport of the frame f v ,...,vk t “ p 1 q along the geodesic cvi with starting vector vi. The precise definition of the frame flows is given below.

k Let f v ,...,vk Tf T M, and let cv be the geodesic on M such that “ p 1 q P i k 1 cv 0 π f and c 0 vi. The i-th frame ﬂow, i 1,...,k, is the map i p q“ p q vi p q“ “

F i : T kM T kM t ÝÑ

f v ,...,vk fv t v v t ,..., vk v t “ p 1 q ÞÑ i p q “ pp 1q i p q p q i p qq where fv t denotes the parallel transport of the frame f along the geodesic cv , i.e. i p q i t every vector vj of f is parallel transported along cv . In particular, vi v t φ vi , i p q i p q“ p q where φt is the geodesic ﬂow on TM (see (1.1.2)). Its inﬁnitesimal generator is given by

i d i d G f Ft f cvi t ;0,..., 0 vi;0,..., 0 , p q“ dt t“0 p q– dt t“0 p q “ p q ˇ ´ ˇ ¯ i ˇ ˇk i.e., G f is a horizontalˇ vector of Tf Tˇ M for all i 1,...,k. p q “ The frame flows act on the frame bundle P kM as well, and their generators are again horizontal vector on P kM. The first frame flow on P kM has been extensively

1 studied in relation to ergodicity. Below, we list the conditions for which Ft is ergodic.

(i) If M is a manifold of odd dimension diﬀerent from 7 with negative curvature [BG80].

(ii) For the set of metrics with negative curvature that is open and dense in the C3 topology [Br75]. 2.3. Diﬀerential Operators 17

(iii) If M is a manifold of even dimension diﬀerent from 8 with pinched negative curvature, pinching constant bigger than 0.93 [BK84].

(iv) If M is a manifold of dimension 7 or 8 with pinched negative curvature, pinching constant bigger than 0.99023... [BP03].

However, in this dissertation, we are not interested in studying the ergodicity of the frame ﬂows further. Our goal is to study a related property, namely, invariance properties of smooth functions on the frame bundle under the action of the frame ﬂows.

2.3 Diﬀerential Operators

As in the classical case of Riemannian manifolds, we have differential operators on T kM and P kM such as the gradient of a smooth function, the covariant derivative and the divergence. However, here we need to distinguish between the horizontal and vertical distribution when defining these operators. In what follows, all inner products are with respect to the metric on M, unless stated otherwise. First, we introduce the notion of semi-basic vector field. We define the pullback bundle π˚ T kM v,f TM T kM π v πk f which is a vector bundle p q “ tp q P ˆ | p q “ p qu over T kM.A semi-basic vector field is an element of X π˚ T kM X : T kM p p qq “ t Ñ k TM smooth X f T k M f T M . | p q P π pfq @ P u k An example of semi-basic vector field is the vector field Vi : T M TM such ÝÑ that f v ,...,vk vi. It will appear extensively in the next chapter. “ p 1 q ÞÑ k k Let ϕ : T M R be a smooth function and let f v ,...,vk T M with ÝÑ “ p 1 q P πk f p. The gradient of ϕ with respect to the metric on T kM is p q“ h v,1 v,k k gradϕ f gradϕ f ; gradϕ f ,..., gradϕ f Tf T M. p q “ p p q p q p qq P

The horizontal and vertical component, called horizontal and i-th vertical gradi- 2.3. Diﬀerential Operators 18

ent, are described intrinsically as follows. Let u TpM, then P h d gradϕ f ,u ϕ fu t , x p q y“ dt t“0 p p qq v,i ˇ d ˇ gradϕ f ,u ϕ v1,...,vˇ i´1,vi tu,vi`1,...,vk , x p q y“ dt t“0 p ` q ˇ ˇ i.e., the horizontal and i-th verticalˇ gradient of ϕ are the derivatives of ϕ along the k horizontal curve t fu t in T M and along the vertical curve t vi tu in the ÞÑ p q ÞÑ ` k i-th TpM copy of T M, respectively. We remind the reader that fu t is the parallel p q k transport of the frame f along the geodesic cu starting at cu 0 π f with initial p q“ p q 1 speed cu 0 u. p q“ h v,i We observe that gradϕ and gradϕ are semi-basic vector ﬁelds.

k k If f P M, gradϕ f defined as above is not an element of Tf P M, as we P p q explained in Subsection 2.1.2. According to (2.1.4), the orthogonal projection of v,i k gradϕ f into Tf P M for f v ,...,vk is p q “ p 1 q v,i v,i 1 k v,i v,j gradoϕ f : gradϕ f gradϕ f ,v gradϕ f ,v v . (2.3.5) 2 j i j p q “ p q´ j“1 x p q y ` x p q y ÿ ´ ¯ h v,1 v,k k Then, gradϕ f ; gradoϕ f ,..., gradoϕ f Tf P M. p p q p q p qq P Let X : T kM TM be a semi-basic vector field. The horizontal and i-th ÝÑ vertical covariant derivative of X with respect to u TpM are given by P h D ∇uX f X fu t , p q“ dt t“0 p p qq v,i ˇ D ˇ ∇uX f X v1,...,vˇ i´1,vi tu,vi`1,...,vk . p q“ dt t“0 p ` q ˇ ˇ As the definitions of theseˇ two operators rely on the usual notion of covariant derivative, they satisfy additivity and the product rule, namely,

¨ ¨ ¨ (i) ∇u X f Y f ∇uX f ∇uY f , p q` p q “ p q` p q ¨ “ ‰ ¨ ¨ (ii) ∇u ϕX f ϕ f ∇uX f gradϕ f X f , p qp q“ p q p q` p q p q for all X,Y semi-basic vector fields and for all ϕ C8 T kM . P p q Finally, we introduce the horizontal and i-th vertical divergence of a semi-basic vector field. They are defined as follows.

h n h v,i n v,i

divX f ∇ei X f ,ei and divX f ∇ei X f ,ei , p q“ i“1x p q y p q“ i“1x p q y ÿ ÿ 2.3. Diﬀerential Operators 19

k where e ,...,en is an orthonormal basis of TpM for p π f . 1 “ p q As a consequence of i and ii above, these operators are additive and satisfy p q p q the product rules

h h h div ϕX f ϕ f divX f gradϕ f ,X f , (2.3.6) p qp q“ p q p q ` x p q p qy v,i v,i v,i div ϕX f ϕ f divX f gradϕ f ,X f , (2.3.7) p qp q“ p q p q ` x p q p qy for all X,Y semi-basic vector ﬁelds and for all ϕ C8 T kM . P p q We conclude the section giving an example of how to calculate the horizontal and i-th vertical covariant derivative and divergence of a very special semi-basic vector ﬁeld, which we will use in the next chapter.

k Example 2.3.1. Let Vi : T M TM be a semi-basic vector ﬁeld such that ÝÑ f v ,...,vk vi. Then, we have the followings. “ p 1 q ÞÑ h D D ∇uVi f Vi fu t vi u t 0, p q“ dt t“0 p p qq “ dt t“0p q p q“ v,j ˇ ˇ ˇ d ˇ ∇uVi fˇ vi δijtuˇ δiju. p q“ dt t“0 ` “ ˇ ˇ Consequently, ˇ

h divVi f 0, (2.3.8) p q“ v,j divVi f nδij. (2.3.9) p q“ Chapter 3

The Pestov Identity on Frame Bundles

In this chapter we state the main results of Part I. In Section 3.1 we present the Lifted Pestov Identity, a Pestov-type identity for smooth functions defined on T kM, whose proof follows the steps of a coordinate free-proof given by Knieper in [Kni02]. In Section 3.2, we integrate the Lifted Pestov Identity over the frame bundle P kM obtaining the Integrated Pestov Identity. More- over, we restrict it to smooth functions defined on P kM and we derive a new identity for smooth functions invariant under one of the frame flows. We remind the reader that all the inner product are with respect to the metric on M, unless specified.

3.1 The Lifted Pestov Identity

We ﬁrst prove some useful relations between the horizontal and vertical diﬀerential operators introduced in the previous chapter.

8 k k k Lemma 3.1.1. Let ϕ C T M , f T M and u,w TpM with p π f . Let P p q P P “ p q i 1,...,k, then “ v,i h h v,i ∇wgradϕ f ,u ∇ugradϕ f ,w . (3.1.1a) x p q y “ x p q y In particular, it follows v,i h h v,i divgradϕ f divgradϕ f . (3.1.1b) p q“ p q 20 3.1. The Lifted Pestov Identity 21

Proof. Using the deﬁnitions of horizontal and i-th vertical covariant derivative and gradient we have v,i h d h ∇wgradϕ f ,u gradϕ v1,...,vi´1,vi tw,vi`1,...,vk ,u x p q y“ dt t“0x p ` q y ˇ B ˇ B ϕ v1 u s ,..., vi u s t w u s ,..., vk u s “ tˇt“0 s s“0 pp q p q p q p q` p q p q p q p qq B ˇ B v,iˇ d ˇ ˇ ˇ gradˇ ϕ fu s ,w “ ds s“0x p p qq y h ˇ v,i ˇ ∇ugradϕ f ,w , “ x ˇ p q y which proves (3.1.1a). Equation (3.1.1b) follows by taking the trace.

8 k k Lemma 3.1.2. Let ϕ C T M , f v ,...,vk T M and u,w TpM with P p q “ p 1 q P P p πk f . Then “ p q h h h h k v,i ∇wgradϕ f ,u ∇ugradϕ f ,w R gradϕ f ,vi w,u , (3.1.2a) x p q y ´ x p q y“ i“1x p p q q y ÿ and k v,l i j j i G G ϕ f G G ϕ f R gradϕ f ,vl vi,vj . (3.1.2b) p q´ p q“ l“1x p p q q y ÿ Proof. We ﬁrst prove (3.1.2a) since (3.1.2b) follows as a consequence.

k Let H t,s fw t s be a variation in T M, i.e., H t,s H1 t,s ,...,Hk t,s p q“ p q uwptq p q p q “ p p q p qq where Hi t,s ` vi w ˘t s . Then, p q“ p q p q uwptqp q ` h h ˘ d h ∇wgradϕ f ,u gradϕ fu t ,uw t x p q y“ dt t“0x p p q p qy ˇ ˇ B B ϕ fw t uwptq s “ tˇt“0 s s“0 p p qq p q B ˇ B ˇ ´ ¯ B ˇ B ˇ ϕ H t,s . “ tˇt“0 sˇs“0 p p qq B ˇ B ˇ ˇ ˇ Now, using (2.1.2), the deﬁnition ofˇ horizontalˇ and i-th vertical component, and the deﬁnition of horizontal covariant derivative, we obtain

h h B B ϕ H t,s B gradϕ H 0,s , B H t,s s s“0 t t“0 p p qq “ s s“0 p p qq t t“0 p q B ˇ B ˇ B ˇ @ ”B ˇ ı D ˇ ˇ ˇ B ˇ pπk˝Hqpt,sq ˇ ˇ ˇ Bt ˇt“0 loooooooomoooooooon k v,i ˇ v,i B gradϕ H 0,sˇ , B H t,s ` s s“0 p p qq t t“0 p q B ˇ i“1 ”B ˇ ı ÿ @ D D ˇ ˇ Hipt,sq ˇ ˇdt t“0 looooooooomooooooooon ˇ ˇ 3.1. The Lifted Pestov Identity 22

D h h D gradϕ H 0,s ,w gradϕ f , B πk H t,s “ xds s“0 p p qq y ` x p q ds s“0 t t“0p ˝ qp qy ˇ ˇ B ˇ D B ˇ ˇ ˇ cu ptqpsq“0 ˇ ˇ dt t“0 Bˇs s“0 w looooooooooooooomooooooooooooooon k D v,i D ˇ v,iˇ D D gradϕ H 0,s , Hi t, 0 ˇ gradˇ ϕ f , Hi t,s ` xds s“0 p p q dt t“0 p qy ` x p q ds s“0 dt t“0 p qy i“1 ˇ ˇ ˇ ˇ ÿ D ˇ ˇ pviqwptq“0 ˇ ˇ ˇ dt ˇt“0 ˇ ˇ looooooomooooooon h h k v,i ˇ D D ∇ugradϕ f ,w gradϕ fˇ , Hi t,s , “ x p q y` x p q ds s“0 dt t“0 p qy i“1 ˇ ˇ ÿ ˇ ˇ where cu t s is the footpoint curve of H t,s . wp qp q ˇ p qˇ Finally,

D D D D Hi t,s Hi t,s ds s“0 dt t“0 p q“ dt t“0 ds s“0 p q ˇ ˇ ˇk ˇ k ˇ ˇ R B πˇ Hiˇ 0,s , B π Hi t, 0 Hi 0, 0 R u,w vi. ˇ ˇ ` s s“0p ˇ ˝ ˇqp q t t“0p ˝ qp q p q“ p q ´B ˇ B ˇ ¯ ˇ ˇ Hence, ˇ ˇ

h h h h k v,i ∇wgradϕ f ,u ∇ugradϕ f ,w R u,w vi, gradϕ f . x p q y ´ x p q y“ i“1x p q p qy ÿ We now prove (3.1.2b). First we observe that

i d i d i G ϕ f ϕ Ft f gradϕ f , Ft f T T kM p q“ dt t“0 p p qq “ p q dt t“0 p q f (3.1.3) ˇ h ˇ ˇ @ ˇ D gradϕ f ,vi . ˇ “ x p q yˇ Therefore,

h h i j i d G G ϕ f G gradϕ f ,vj gradϕ fvi t , vj vi t p q“ x p q y“ dt t“0x p p qq p q p qy ˇ h h h Dˇ ˇ gradϕ fvi t ,vj ∇vi gradϕ f ,vj , “ xdt t“0 p p qq y “ x p q y ˇ h h ˇ j i and G G ϕ f ∇v gradϕ f ,vi . ˇ p q “ x j p q y Choosing u vj and w vi in (3.1.2a), we obtain (3.1.2b). “ “ 8 k k Lemma 3.1.3. Let ϕ C T M , f v ,...,vk and w TpM with p π f . P p q “ p 1 q P “ p q Then,

h h k v,l j j gradG ϕ f ,w G gradϕ f ,w R gradϕ f ,vl w,vj , (3.1.4a) x p q y“ x p q y` l“1x p p q q y v,i v,i ÿ h j j gradG ϕ f ,w G gradϕ f ,w δij gradϕ f ,w . (3.1.4b) x p q y“ x p q y` x p q y 3.1. The Lifted Pestov Identity 23

In particular,

v,i v,i j j l gradG ϕ f ,vl G gradϕ f ,vl δijG ϕ f . (3.1.4c) x p q y“ x p q y` p q Proof. We ﬁrst prove (3.1.4a). Using (3.1.3) and (3.1.2a) we obtain,

h j d j gradG ϕ f ,w G ϕ fw t x p q y“ dt t“0 p p qq ˇ h d ˇ ˇ gradϕ fw t , vj w t “ dt t“0x p p qq p q p qy hˇ h ˇ ∇wgradϕ f ,vj “ x ˇ p q y h h k v,l

∇vj gradϕ f ,w R gradϕ f ,vl w,vj “ x p q y` l“1x p p q q y ÿ d h k v,l gradϕ fvj t , w vj t R gradϕ f ,vl w,vj dt t“0 “ x p p qq p q p qy ` l“1x p p q q y ˇ ÿ ˇ h k v,l jˇ G gradϕ f ,w R gradϕ f ,vl w,vj . “ x p q y` l“1x p p q q y ÿ We now prove (3.1.4b). Using (3.1.3) and (3.1.1a), we have

v,i j d j gradG ϕ f ,w G ϕ v1,...,vi tw,...,vk x p q y“ dt t“0 p ` q ˇ h d ˇ ˇ gradϕ v1,...,vi tw,...,vk ,vj δijtw “ dt t“0x p ` q ` y v,iˇ h h ˇ ∇wgradϕ f ,vj δij gradϕ f ,w “ x ˇ p q y` x p q y h v,i h ∇v gradϕ f ,w δij gradϕ f ,w “ x j p q y` x p q y d v,i h gradϕ fvj t , w vj t δij gradϕ f ,w “ dt t“0x p p qq p q p qy ` x p q y ˇ v,i h jˇ G gradϕ f ,w δij gradϕ f ,w . “ ˇx p q y` x p q y Setting w vl, in the last equality above, the second term of the RHS is h “ l gradϕ f ,vl G ϕ f by (3.1.3), and we obtain (3.1.4c). x p q y“ p q We are now ready to prove the Lifted Pestov Identity. We recall that the proof can be compared with [Kni02, Theorem 1.1 in Appendix].

Theorem 3.1.4 (Lifted Pestov Identity). Let ϕ C8 T kM , Then, P p q v,j h h i j,i 2 divZ f divY f δij gradϕ f p q` p q` } p q} “ k v,l v,j h v,j i R gradϕ f ,vl vi, gradϕ f 2 gradϕ f , gradG ϕ f , (3.1.5) “ l“1x p p q q p qy ` x p q p qy ÿ 3.1. The Lifted Pestov Identity 24 where h v,j v,j j,i i Y f gradϕ f , gradϕ f vi G ϕ f gradϕ f , p q “ x p q p qy ´ p q p q and ` ˘ h Zi f Giϕ f gradϕ f , p q“ p q p q for all i, j 1,...,k. ` ˘ “ Proof. Using equations (2.3.7) and (3.1.3) and the deﬁnitions of horizontal and j-th vertical gradient and covariant derivative, we have

v,j v,j h divZi f div Giϕ f gradϕ f p q“ p q p q v,j h v,j h Giϕ f div` gradϕ ˘f gradGiϕ f , gradϕ f “ p q p q ` x p q p qy h d i dt G ϕpv1,...,vj `tgradϕpfq,...,vkq loooooooooooooomoooooooooooooont“0 v,j h i ˇ G ϕ f divgradϕ f ˇ “ p q p q d h h h gradϕ v1,...,vj tgradϕ f ,...,vk ,vi δijtgradϕ f ` dt t“0x p ` p q q ` p qy v,j ˇ h v,j h h i ˇ 2 G ϕ f divgradˇ ϕ f ∇ h gradϕ f ,vi δij gradϕ f . “ p q p q ` x gradϕpfq p q y` } p q}

In the same way, using (2.3.6) in the second equality and (3.1.1b) in the third equality, we obtain

h h h v,j h v,j j,i i divY f div gradϕ f , gradϕ f vi div G ϕ f gradϕ f p q“ px p q p qy q´ p p q p qq h v,j h h h v,j gradϕ f , gradϕ f divVi f grad gradϕ f , gradϕ f ,vi “ x p q p qy p q `x px p q p qyq y “0 by (2.3.8) h v,j looomooonh v,j Giϕ f divgradϕ f gradGiϕ f , gradϕ f ´ p q p q ´ x p q p qy d Giϕ f ptq dt t“0 v,j loooooooooooooomoooooooooooooongradϕpfq d h v,j ˇ ` v,j˘ h ˇ i gradϕ fvi t , gradϕ fvi t G ϕ f divgradϕ f “ dt t“0x p p qq p p qqy ´ p q p q ˇ h ˇ d ˇ gradϕ f v,j t , vi v,j t ´ dt t“0x p gradϕpfqp q p qgradϕpfqp qy ˇ h h ˇ v,j h h v,j ∇v gradϕ f ˇ, gradϕ f gradϕ f , ∇v gradϕ f “ x i p q p qy ` x p q i p qy v,j h h h i G ϕ f divgradϕ f ∇ v,j gradϕ f ,vi . ´ p q p q ´ x gradϕpfq p q y 3.2. The Integrated Pestov Identity 25

Hence, summing these two divergences and using (3.1.1a) in the second equality and (3.1.2a) in the third equality, we have

v,j h h h v,j h h i j,i divZ f divY f ∇vi gradϕ f , gradϕ f ∇ v,j gradϕ f ,vi p q` p q “ x p q p qy ´ x gradϕpfq p q y h h v,j v,j h h 2 gradϕ f , ∇vi gradϕ f ∇ h gradϕ f ,vi δij gradϕ f ` rx p q p qy ` x gradϕpfq p q ys ` } p q} h h v,j h h

∇vi gradϕ f , gradϕ f ∇ v,j gradϕ f ,vi “ x p q p qy ´ x gradϕpfq p q y h v,j h h 2 2 ∇v gradϕ f , gradϕ f δij gradϕ f ` x i p q p qy ` } p q} h h v,j h k v,l v,j 2 δij gradϕ f 2 ∇vi gradϕ f , gradϕ f R gradϕ f ,vl vi, gradϕ f . “ } p q} ` x p q p qy ` l“1x p p q q p qy ÿ Finally, using the deﬁnition of horizontal and vertical gradient, we have

h v,j h v,j h i 2 gradϕ f , gradG ϕ f 2 gradϕ f , grad gradϕ,Vi f x p q p qy “ x p q px yqp qy d d h 2 ϕ v1 vi s ,..., vj t gradϕ f vi s ,..., vk vi s “ ds s“0 dt t“0 p q p q p ` p p qq p qq p q p q ˇ ˇ h ˇ ˇ ` ˘ 2 δij gradϕ f ˇ ˇ ` } p q} d h h h 2 gradϕ v1,...,vj tgradϕ f ,...,vk ,vi δijtgradϕ f “ dt t“0x p ` p q q ` p qy hˇ v,j h h ˇ 2 2 ∇v gradϕ f , gradϕ f 2δij gradϕ f . “ x ˇ i p q p qy ` } p q} Substituting this expression in the previous identity we obtain the theorem.

3.2 The Integrated Pestov Identity

We now prove the integrated version of (3.1.5) over the frame bundle P kM with respect to the measure dµp f dvol p dνp f where dν is the measure on the ﬁbre p q“ p q p q k Fp of P M and dvol is the measure on M. Before integrating, we show the behaviour of the horizontal divergence and of the generators of the frame ﬂows under integration.

Lemma 3.2.1. Let X : T kM TM be a semi-basic vector ﬁeld, then ÝÑ h h divX f dµ f divX f dvol p dνp f 0. (3.2.6) k p q p q“ p q p q p q“ żP M żM żFp 3.2. The Integrated Pestov Identity 26

Proof. We ﬁrst recall that Fp O n O n k . To prove the Lemma, it suﬃces to – p q{ p ´ q show h ∇vX,v dνp f ∇v X f dνp f ,v . (3.2.7) x y p q “ x p q p q y żFp żFp In fact, from the above it follows

h divX f dνp f div X f dνp f . p q p q“ p q p q żOpnq{Opn´kq żOpnq{Opn´kq Then, due to the compactness of M we obtain (3.2.6). Let us prove (3.2.7).

n k We define ̺ : P M P M with ̺ v ,...,vn v ,...,vk . In particular, Ñ pp 1 qq “ p 1 q n ̺ : F˜p Fp where F˜p O n is the fibre of the principal bundle P M with O n Ñ – p q p q right action. On the fibre O n O n k there exists a unique (up to scalar) left O n -invariant p q{ p ´ q p q Haar measure (see, e.g., [Fer98, Thereom 8.1.8]) obtained in the following way.

Fix f F˜p and let g O n , Then, there exists ψf : O n O n k Fp such 0 P P p q 0 p q{ p ´ q Ñ ˚ that g O n k ̺ g f , canonical diﬀeomorphism. The pullback ψ νp ¨ p ´ q ÞÑ p ¨ 0q f0 p q “ θO n O n k gives the measure on O n O n k . p q{ p ´ q p q{ p ´ q In what follows, we drop the arguments of the measure and the subscript of θ to ease the notation. We have,

h D ∇vX f ,v dνp f X fv t ,v dνp f F x p q y p q“ F xdt t“0 p p qq y p q ż p ż p ˇ ˇ d ˇ X fv t ,v dνp f . “ F dt t“0x p p qq y p q ż p ˇ ˇ Note that for f˜ g f F˜p, ˇ “ ¨ 0 P

X fv t X ̺ f˜v t X ̺ g f v t X ψ f t g . p p qq “ p p p qqq “ p p ¨ p 0q p qqq “ p p 0qvp qp qq Then,

d d t X fv t ,v dνp f X fv t ,φ v dνp f F dt t“0x p p qq y p q“ ψ Opnq{Opn´kq dt t“0x p p qq p qy p q ż p ˇ ż f0 ˇ ˇ d ˇ ˇ ` ˘t ˇ ˚ X ψpf0qvptq g ,φ v d ψf0 νp g “ Opnq{Opn´kq dt t“0x p p qq p qy p qp q ż ˇ d ˇ t ˇ X ψpf0qvptq g ,φ v dθ g O n k “ dt t“0 Opnq{Opn´kqx p p qq p qy p ¨ p ´ qq ˇ ż ˇ ˇ 3.2. The Integrated Pestov Identity 27

d t X ψpf0qvptq g dθ,φ v “ dt t“0x Opnq{Opn´kq p p qq p q y ˇ ż Dˇ ˇ X ψf0 g dθ,v “ xdt t“0 Opnq{Opn´kq p p qq y ˇ ż D ˇ ˇ X f dνp f ,v , “ xdt t“0 F k p q p q y ˇ ż p ˇ which concludes the proof.ˇ

Lemma 3.2.2. Let ϕ,ψ C8 T kM , then P p q

ψ f Giϕ f dµ ϕ f Giψ f dµ. (3.2.8) k p q p q “´ k p q p q żP M żP M h h i Proof. We observe that G ϕ f gradϕ f ,vi div ϕVi f by equations (3.1.3), p q “ x p q y“ p qp q (2.3.6) and (2.3.8). Therefore, Giϕ f dµ 0 by Lemma 3.2.1. This fact and P kM p q “ the Leibniz rule prove the Lemma.ş

We are now ready to state the integrated version of (3.1.5).

Theorem 3.2.3 (Integrated Pestov Identity). Let ϕ C8 T kM , then P p q

h k v,l v,j 2 δij gradϕ L2pP kMq R gradϕ f ,vl vi, gradϕ f dµ } } ´ k x p p q q p qy “ żP M l“1 h ÿ v,j h v,j gradϕ f , gradGiϕ f gradGiϕ f , gradϕ f dµ. (3.2.9) “ k x p q p qy ` x p q p qy żP M Proof. Consider equation (3.1.5). Under integration over P kM the horizontal divergence vanishes and the remaining non-zero terms are

v,j h h v,j i 2 i divZ f dµ δij gradϕ f L2pP kMq 2 gradϕ f , gradG ϕ f dµ k p q ` } p q} “ k x p q p qy żP M żP M k v,l v,j R gradϕ f ,vlvi, gradϕ f dµ. (3.2.10) k ` P M l“1x p p q p qy ż ÿ We have

v,j v,j h divZi f dµ div Giϕ f gradϕ f dµ k p q “ k p p q p qq żP M żP M h v,j v,j h gradϕ f , gradGiϕ f Giϕ f divgradϕ f dµ “ k x p q p qy ` p q p q żP M h v,j h v,j gradϕ f , gradGiϕ f Giϕ f divgradϕ f dµ “ k x p q p qy ` p q p q żP M 3.2. The Integrated Pestov Identity 28

h v,j h v,j gradϕ f , gradGiϕ f gradGiϕ f , gradϕ f dµ “ k x p q p qy ´ x p q p qy ` żP M h v,j div Giϕ f gradϕ f dµ ` k p p q p qq żP M h v,j h v,j gradϕ f , gradGiϕ f gradGiϕ f , gradϕ f dµ. “ k x p q p qy ´ x p q p qy żP M Substituting into (3.2.10) we have the theorem.

3.2.1 Restriction to functions on the frame bundle P kM

Theorem 3.2.3 can be restricted to smooth functions on P kM. Below, we present its formulation.

8 k k Theorem 3.2.4. Let ϕ C P M and f v ,...,vk P M. Then P p q “ p 1 q P

h k k v,j v,i 2 k 1 i 2 k gradϕ L2 ` G ϕ L2 R gradoϕ f ,vj vi, gradoϕ f dµ 2 k } } ´ i“1 } } ´ P M i,j“1x p p q q p qy “ ÿ ż ÿ k h v,i h v,i i i gradϕ f , gradoG ϕ f gradG ϕ f , gradoϕ f dµ, (3.2.11) k “ i“1 P M x p q p qy ` x p q p qy ÿ ż where L2 stands for the L2-space on P kM.

Proof. Letϕ ˜ be a smooth extension of ϕ on T kM and consider equation (3.2.9). Setting j i and summing over i 1,...,k we obtain “ “

h k v,l v,i 2 k grad˜ϕ L2 R grad˜ϕ f ,vl vi, grad˜ϕ f dµ k } } ´ P M i,l“1x p p q q p qy “ ż ÿ k h v,i h v,i grad˜ϕ f , gradGiϕ˜ f gradGiϕ˜ f , grad˜ϕ f dµ. (3.2.12) k “ P M i“1x p q p qy ` x p q p qy ż ÿ We consider the RHS and using equations (2.3.5) and (3.1.3) we obtain

h v,i h v,i i i grad˜ϕ f , gradG ϕ˜ f dµ grad˜ϕ f , gradoG ϕ˜ f dµ k x p q p qy “ k x p q p qy żP M żP M k v,i v,j 1 j i j i G ϕ˜ f gradG ϕ˜ f ,vj G ϕ˜ f gradG ϕ˜ f ,vi dµ, (3.2.13) 2 k ` j“1 P M p qx p q y` p qx p q y ÿ ż 3.2. The Integrated Pestov Identity 29 and

v,i h v,i h i i grad˜ϕ f , gradG ϕ˜ f dµ gradoϕ˜ f , gradG ϕ˜ f dµ k x p q p qy “ k x p q p qy żP M żP M k v,j v,i 1 j i j i G G ϕ˜ f grad˜ϕ f ,vi G G ϕ˜ f grad˜ϕ f ,vj dµ 2 k ` j“1 P M p qx p q y` p qx p q y ÿ ż v,i h i k 1 i 2 gradoϕ˜ f , gradG ϕ˜ f dµ p ` q G ϕ˜ f L2 “ k x p q p qy ` 2 } p q} żP M k v,i v,j 1 i j i j G ϕ˜ f gradG ϕ˜ f ,vj G ϕ˜ f gradG ϕ˜ f ,vi dµ, 2 k ´ j“1 P M p qx p q y` p qx p q y ÿ ż (3.2.14) where we used Lemma 3.2.2 and 3.1.3 for the second equality. Adding (3.2.13) and (3.2.14) and taking the sum over i, the RHS of (3.2.12) becomes

k h v,i h v,i i i grad˜ϕ f , gradoG ϕ˜ f gradG ϕ˜ f , gradoϕ˜ f dµ k x p q p qy ` x p q p qy żP M i“1 ÿ k k 1 i 2 ` G ϕ˜ 2 . (3.2.15) 2 L ` i“1 } } ÿ Finally, using (2.3.5) and the antisymmetry of R, we have

k v,j v,i k v,j v,i R gradϕ f ,vj vi, gradϕ f R gradoϕ f ,vj vi, gradoϕ f . (3.2.16) i,j“1x p p q q p qy “ i,j“1x p p q q p qy ÿ ÿ Substituting (3.2.15) and (3.2.16) into (3.2.12) proves the theorem.

The form of the previous theorem allows us to derive a new identity assuming the function ϕ to be invariant under one the frame flows. The new formula contains only three terms, the Riemannian curvature tensor, the L2 norms of the generators of the frame flows, and the L2 norm of the horizontal gradient. Therefore, it shows a close connections between curvature properties of the manifold and properties of the frame flows.

Corollary 3.2.5. Let ϕ C8 P kM and assume it is invariant under the i-th frame P p q i k k ﬂow, i.e., G ϕ f 0 for all f P M, and let f v ,...,vk P M. Then p q“ P “ p 1 q P k n´k h n 1 j 2 G ϕ L2 2 grad˜ϕ f ,el L2 R wj,vj vi,wi dµ, (3.2.17) 2 n j“1,j‰i } } ` l“1 }x p q y} “ j“1 P M x p q y ÿ ÿ ÿ ż 3.2. The Integrated Pestov Identity 30

v,j where v ,...,vk,e ,...,en k is an orthonormal basis of T k M, wj gradoϕ f , p 1 1 ´ q π pfq “ p q and L2 stands for the L2-space on P kM.

Proof. We prove the theorem for i 1. The other cases follow in the same way. “ n n Let f v ,...,vn P M andϕ ˜ a smooth extension of ϕ on T M and let “ p 1 q P k v ,...,vk,e ,...,en k be an orthonormal basis of TpM, p π f . p 1 1 ´ q “ p q We ﬁrst observe that due to equation (2.3.5), we have

v,i v,i gradoϕ˜ f ,el grad˜ϕ f ,el , l 1,...,n k. (3.2.18) x p q y “ x p q y @ “ ´

We consider equation (3.2.11) and we aim at rewriting the horizontal gradient and its RHS in terms of L2-norms of the generators of the frame ﬂows and in terms of the Riemannian curvature tensor. First of all, we observe that

h n n´k h i grad˜ϕ f G ϕ˜ f vi grad˜ϕ f ,el el, (3.2.19a) p q“ i“1 p q ` l“1 x p q y ÿ ÿ h k n´k h 2 i 2 2 grad˜ϕ L2 G ϕ˜ L2 grad˜ϕ f ,el L2 . (3.2.19b) } } “ i“1 } } ` l“1 }x p q y} ÿ ÿ

We now look at the RHS of (3.2.11). From (2.3.5), we derive

1 v,i v,j wi,vj grad˜ϕ f ,vj grad˜ϕ f ,vi , j 1,...k. (3.2.20) x y“ 2 x p q y ´ x p q y @ “

` v,i ˘ wi,el grad˜ϕ f ,el , l 1,...,n k. (3.2.21) x y “ x p q y @ “ ´ 3.2. The Integrated Pestov Identity 31

Using these equations, Lemma 3.1.3 and equation (3.2.19a), we have

k h v,i k k v,i i j i grad˜ϕ f , gradoG ϕ˜ f dµ G ϕ˜ f gradoG ϕ˜ f ,vj dµ k k i“2 P M x p q p qy “ i“2 P M j“2,j‰i p qx p q y ÿ ż ÿ ” ż ÿ n´k h v,i i grad˜ϕ f ,el gradG ϕ˜ f ,el dµ k ` P M l“1 x p q yx p q y ż ÿ ı k v,i v,j 1 j i i G ϕ˜ f gradG ϕ˜ f ,vj gradG ϕ˜ f ,vi dµ k 2 “ i,j“2,j‰i P M p q x p q y ´ x p q y ÿ ż ” ı k n´k h h v,i i grad˜ϕ f ,el grad˜ϕ f ,el G grad˜ϕ f ,el dµ k ` i“2 l“1 ` P M x p q y x p q y` px p q yq ÿ ÿ ż ” ı k v,i v,j 1 j i i j G ϕ˜ f G grad˜ϕ f ,vj G grad˜ϕ f ,vi G ϕ˜ f dµ k 2 “ i,j“2,j‰i P M p q x p q y´ x p q y` p q ÿ ż ” ı k n´k h v,i n´k h i 2 grad˜ϕ f ,el G grad˜ϕ f ,el dµ k 2 grad˜ϕ f ,el L2 k ` i“2 l“1 P M x p q y px p q yq ` p ´ q l“1 }x p q y} ÿ ÿ ż ÿ

k k i j k 2 i 2 G G ϕ˜ f wi,vj dµ ´ G ϕ˜ L2 k 2 “ i,j“2,j‰i P M ´ p qx y ` i“2 } } ÿ ż ÿ k n´k h v,i n´k h i 2 grad˜ϕ f ,el G grad˜ϕ f ,el dµ k 2 grad˜ϕ f ,el L2 , k ` i“2 l“1 P M x p q y px p q yq ` p ´ q l“1 }x p q y} ÿ ÿ ż ÿ (3.2.22) and

k h v,i k k h v,i i i gradG ϕ f , gradoϕ f dµ gradG ϕ˜ f ,vj vj, gradoϕ˜ f dµ k k i“2 P M x p q p qy “ i“2 j“1,j‰i P M x p q yx p qy ÿ ż ÿ ÿ ż k n´k h v,i i gradG ϕ˜ f ,el el, gradoϕ˜ f dµ ` k x p q yx p qy i“2 l“1 żP M k ÿ ÿ k 1 i j i G G ϕ˜ f v1,wi dµ G G ϕ˜ f wi,vj dµ k k “ i“2 P M p qx y ` i,j“2,j‰i P M p qx y ÿ ż ÿ ż k n´k h v,i i G grad˜ϕ f ,el grad˜ϕ f ,el dµ k ` i“2 l“1 P M px p q yqx p q y ÿ ÿ ż k n´k k v,j v,i R grad˜ϕ f ,vj vi,el grad˜ϕ f ,el dµ. (3.2.23) k ´ i“2 l“1 P M j“1x p p q q yx p q y ÿ ÿ ż ÿ Summing (3.2.22) and (3.2.23), using (3.1.2b), (3.2.8) and the skew-symmetry of the matrix wi,vj i,j, we have px yq 3.2. The Integrated Pestov Identity 32

k h v,i h v,i i i gradϕ f , gradoG ϕ f gradG ϕ f , gradoϕ f dµ k i“1 P M x p q p qy ` x p q p qy “ ÿ ż k n´k h k 2 i 2 2 ´ G ϕ˜ 2 k 2 grad˜ϕ f ,el 2 2 L L “ i“2 } } ` p ´ q l“1 }x p q y} k ÿ ÿ k 1 i j i i j G G ϕ˜ f v1,wi dµ G G G G ϕ˜ f vj,wi dµ k k ` i“2 P M p qx y ` i,j“2,i‰j P M p ´ q p qx y ÿ ż ÿ ż k n´k k v,j v,i R grad˜ϕ f ,vj vi,el grad˜ϕ f ,el dµ k ´ i“2 l“1 P M j“1x p p q q yx p q y ÿ ÿ ż ÿ k n´k h v,i h v,i i i grad˜ϕ f ,el G grad˜ϕ f ,el G grad˜ϕ f ,el grad˜ϕ f ,el dµ k ` i“2 l“1 P M x p q y px p q yq ` px p q yqx p q y ÿ ÿ ż h v,i i PkM G pxgradϕ ˜pfq,elyxgradϕ ˜pfq,elyq dµ“0 (see proof of Lemma 3.2.2) loooooooooooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooooooooooon k ş n´k h k k k 2 i 2 2 ´ G ϕ˜ L2 k 2 grad˜ϕ f ,el L2 R wi,vl vi,wi dµ. 2 k “ i“2 } } ` p ´ q l“1 }x p q y} ´ P M i“2 l“1x p q y ÿ ÿ ż ÿ ÿ (3.2.24)

Substituting (3.2.19b) and (3.2.24) into (3.2.11) concludes the proof.

For k n ,the above corollary simpliﬁes as follows. “ Corollary 3.2.6. Let ϕ C8 P nM and assume it is invariant under the i-th frame P p q i n n ﬂow, i.e., G ϕ f 0 for all f P M, and let f v ,...,vn P M. Then p q“ P “ p 1 q P n n 1 j 2 G ϕ L2 R wj,vj vi,wi dµ, (3.2.25) 2 n j“1,j‰i } } “ j“1 P M x p q y ÿ ÿ ż v,j 2 2 n where wj gradoϕ f and L stands for the L -space on P M. “ p q Chapter 4

Dynamical Applications

In this chapter we present two dynamical consequences of Corollary 3.2.6 under curvature assumptions on the base manifold M. In Section 4.1 we consider the principal bundle over a n-dimensional compact manifold with non-positive constant curvature, and over a 2-dimensional manifold of non-positive curvature proving that any smooth functions invariant under one of the frame ﬂows is also invariant under the remaining ones. In Section 4.2 we consider the Grassmannians of oriented k-planes over M with non-positive curvature operator and we show an invariance property of smooth functions on such Grassmannians under the action of the parallel transport.

4.1 An invariance property of smooth functions on frame bundles

Corollary 3.2.6 in Chapter 3 hints that if the curvature of the manifold could be non- positive, then function ϕ would also be invariant under the remaining frame ﬂows. This is true in two cases, for 2-dimensional manifolds of non-positive curvature and for higher dimensional manifolds of non-positive constant curvature.

Corollary 4.1.1. Let M be a 2-dimensional manifold of non-positive curvature. Let ϕ be a smooth function on the principal bundle P 2M. If ϕ is invariant under the i-th frame ﬂow, 0 i 2, i.e., Giϕ f 0 for all f C8 P 2M , then it is invariant ă ď p q “ P p q under the remaining frame ﬂow.

33 4.2. An invariance property of smooth functions on oriented Grassmannians 34 Proof. Consider equation (3.2.25) for k 2 and assume i 1, the case i 2 being “ “ “ the same. Writing the vertical gradients in the RHS according to the orthonormal basis v1,v2, we obtain

v,1 1 2 2 2 G ϕ L2 R v2,v1 v1,v2 gradoϕ f ,v2 dµ 0. 2} } “ 2 x p q yx p q y ď żP M 2 2 2 Hence, G ϕ 2 0 and therefore ϕ is also invariant under G . } }L “ Corollary 4.1.2. Let M be a n-dimensional manifold of non-positive constant sectional curvature K. Let ϕ be a smooth function on the principal bundle P nM. If ϕ is invariant under one of the i-th frame ﬂows, i.e., Giϕ f 0 for all f C8 P nM , p q “ P p q then it is invariant under the remaining frame ﬂows.

Proof. We ﬁrst notice that in case of constant sectional curvature K, the Riemannian curvature tensor decomposes as

R v ,v v ,v K v ,v v ,v v ,v v ,v . x p 1 2q 3 4y“ x 1 4yx 2 3y ´ x 1 3yx 2 4y ` ˘ Substituting this expression into the RHS of (3.2.25) we obtain

n n 1 j G ϕ L2 K wj,wi wj,vi wi,vj dµ K wi L2 0. 2 } } “ n x y ´ x yx y “ } } ď j“1,j‰i żP M j“1 ÿ ÿ ` ˘ n j Hence, G ϕ 2 0. This implies that each term of the sum is zero as j“1,j‰i } }L “ they must allř be non-negative. Therefore, ϕ is invariant under the remaining frame ﬂows.

4.2 An invariance property of smooth functions on oriented Grassmannians

Another consequence of Corollary 3.2.6 is for smooth functions on oriented Grass- mannians. We begin explaining the relation between oriented Grassmannians and frame bundles. 4.2. An invariance property of smooth functions on oriented Grassmannians 35

4.2.1 Oriented Grassmannians and Parallel Transport

The oriented k-th Grassmannian of M, Gk M for 1 k n dim M, is the union orp q ď ď “ of all k-planes of TpM for all p M together with an intrinsic orientation, i.e., P k Gor M Aor A TpM, dim A k , p q“ pPM Ă “ ď ˇ ( where Aor is a subspace of TpM with anˇ additional choice of intrinsic orientation. This is a 2-fold of the non-oriented k-th Grassmannian. We observe that for k 1 “ we have G1 M SM, the unit tangent bundle of M. orp q“ Gk M is linked to the principal bundle P nM by the canonical projection orp q π˜ : P nM Gk M ÝÑ orp q

f v ,...,vn span v ,...,vk , v ,...,vk , “ p 1 q ÞÑ t 1 u p 1 q ` ˘ where v ,...,vk stands for intrinsic orientation. p 1 q The existence of this projection implies that any function ϕ C8 Gk M can P p orp qq be extended to a function φ C8 P nM by setting φ ϕ π˜. The function φ has P p q “ ˝ two important properties. Firstly, since ϕ is invariant under the action of SO k , φ is p q SO k 0 invariant under the action of matrices of the form p q . Secondly, ¨ 0 SO n k ˛ p ´ q its vertical gradients satisfy the following conditions.˝ ‚

Lemma 4.2.1. Let ϕ C8 Gk M and φ ϕ π˜ C8 P nM . Then, P p orp qq “ ˝ P p q v,i (i) gradoφ f span v ,...,vk if i k 1; p q P t 1 u ě ` v,i (ii) gradoφ f span vk ,...,vn if i 1,...k. p q P t `1 u “ v,i Proof. To prove i and ii , we need to show that gradoφ f ,vj 0 for i, j k 1 v,i p q p q x p q y“ ě ` and gradoφ f ,vj 0 for i, j k, respectively. x p q y“ ď Let φ˜ be a smooth extension of φ on T nM constructed as follows.

n Let h : T M 0, be such that h w ,...,wn det wi,wj ij and let ÝÑ r 8s p 1 q “ px yq n ψ : w ,...,wn lin. indep P M be the Gram-Schmidt process. We define the tp 1 q u ÝÑ cut off function H : 0, 0, 1 be a cut off function such that H x 0 for r 8s ÝÑ r s p q “ x 1 , H x 1 for x 3 and 0 H x 1 for 1 x 3 . Then ď 2 p q“ ě 4 ď p qď 2 ď ď 4

H h w1,...,wn φ ψ w1,...,wn w1,...,wn lin. indep. φ˜ w1,...,wn p q p ˝ qp q t u p q“ $ 0 otherwise & ` ˘ % 4.2. An invariance property of smooth functions on oriented Grassmannians 36

Note that φ˜ w ,...,wl twj,...,wn φ˜ w ,...,wn for all l k 1. p 1 ` q“ p 1 q ě ` This implies that, for i, j k 1, we have ě ` v,i 1 v,i 1 v,j gradoφ f ,vj gradφ˜ f ,vj gradφ˜ f ,vi x p q y“ 2x p q y´ 2x p q y 1 d φ˜ v1,...,vk,...,vi tvj,...,vn φ˜ v1,...,vk,...,vj tvi,...,vn 0, “ 2 dt t“0 p ` q´ p ` q “ ˇ ´ ¯ which provesˇ i . ˇ p q Part ii follows from the fact that φ˜ depends only on the plane spanned by the p q ﬁrst k vectors.

Another consequence of the existence ofπ ˜ is a correspondence between the generators of the frame ﬂows on P nM and the parallel transport on Gk M along special orp q directions.

8 n 8 k k In fact, let φ ϕ π˜ C P M with ϕ C G M , f v ,...,vn P M “ ˝ P p q P p orp qq “ p 1 q P and Aor span v ,...,vk , v ,...,vk π˜ f , then “ t 1 u p 1 q “ p q i ` i d ˘ d G φ f G ϕ π˜ f ϕ π˜ fvi t ϕ Aor vi t . (4.2.1) p q“ p ˝ qp q“ dt t“0p ˝ qp p qq “ dt t“0 pp q p qq ˇ ˇ Deﬁnition 4.2.2. Let Aor v t denoteˇ the parallel transportˇ of the oriented k-plane p q p q ˇ ˇ k 1 Aor G M along a curve cv on M with c 0 v. The parallel transport is called P orp q vp q“ intrinsic if the vector v belongs to Aor, and non-intrinsic otherwise.

Aor Aor

Figure 4.1: From left to right: example of intrinsic and non-intrinsic parallel transport of a plane Aor along the geodesic cv.

The above deﬁnition and equation (4.2.1) implies the following. 4.2. An invariance property of smooth functions on oriented Grassmannians 37

Corollary 4.2.3. Let π˜ : P nM Gk M be the canonical projection, ϕ C8 Gk M , ÝÑ orp q P p orp qq and φ ϕ π˜ C8 P nM . Then, ϕ is invariant under all intrinsic parallel transports “ ˝ P p q if and only if φ is invariant under the i-th frame ﬂows, for i 1,...,k. “

4.2.2 Invariance Property

We now assume that M has non-positive curvature operator R. The curvature operator is a linear operator R :Λ2 TM Λ2 TM deﬁned as p q ÝÑ p q

R X Y ,Z W 2 R X,Y W, Z TM (4.2.2) x p ^ q ^ yΛ pTMq “ x p q y for all vector ﬁelds X,Y,Z,W on M. The curvature operator R is symmetric and we say that R is non-positive (R 0) ď if all of its real eigenvalues are non-positive. A manifold M with non-positive curvature operator has non-positive curvature. However, the inverse implication is not true, for details on this topic we refer the reader to [AF04] and [Ara10].

Theorem 4.2.4. Let M be a compact n-dimensional Riemannian manifold with non- positive curvature operator (R 0). Let 1 k n and ϕ C8 Gk M . If ϕ is ď ď ď P p orp qq invariant under all the intrinsic parallel transports then it is also invariant under all parallel transports.

Proof. Let φ ϕ π˜ C8 P nM . “ ˝ P p q Since ϕ is invariant under intrinsic parallel transports, φ is invariant under G1,...,Gk due to (4.2.1). Considering equation (3.2.25) and summing over i 1,...,k we obtain “ n k n k j 2 G φ L2pP nMq R wj,vj vi,wi dµ 2 n j“1 } } “ P M i“1 j“1x p q y ÿ ż ÿ ÿ (4.2.3) n k

R wj vj , wi vi Λ2pTMq dµ, n “ P M x j“1 ^ i“1 ^ y ż ´ ÿ ¯ ÿ v,i where wi gradoφ f . “ p q 4.2. An invariance property of smooth functions on oriented Grassmannians 38

Since the matrix wi,vj is skew-symmetric and making use of Lemma 4.2.1, x y i,j we obtain ` ˘ n n n k n n k

wj vj wj,vl vl vi wj,vl vl vj wj,vl vl vj j“1 ^ “ j“1 l“1,l‰jx y ^ “ j“1 l“k`1x y ^ ` j“k`1 l“1x y ^ ÿ ÿ ÿ ÿ ÿ ÿ ÿ k

2 wj vj. “ j“1 ^ ÿ Since R 0, the RHS of (4.2.3) is non-positive forcing the LHS to be zero. We ď conclude that φ is invariant under all frame ﬂows, and so ϕ is invariant under all parallel transports.

For k 1 the theorem above allows us to recover an unpublished result of Knieper “ [Kni] on the geodesic ﬂow. In fact, G1 M SM, the intrinsic parallel transport orp q “ corresponds to the geodesic ﬂow and the assumption on the non-positivity of the curvature operator can be weakened to the non-positivity of the sectional curvature, as it is clear from the proof of Theorem 4.2.4. Therefore, we can state the following.

Corollary 4.2.5. Let M be a compact Riemannian manifold with non-positive curvature. Let ϕ C8 SM invariant under the geodesic ﬂow, then ϕ is also invariant P p q under parallel transport.

In view of Theorem 4.2.4, it is natural to investigate density properties of orbits of k-planes Aor under all intrinsic parallel transports. In fact, let I Aor be the set p q of all k-planes obtained by finitely many moves along intrinsic parallel transport and G Aor be the set of all k-planes obtained by finitely many moves along general p q parallel transport. Theorem 4.2.4 suggests that even though there might be many k-planes Aor such that I Aor is much smaller and not dense in G Aor , there might p q p q 1 1 1 always be a k-plane A arbitrarily close to Aor such that I A is dense in G A . or p orq p orq This is at least true in the case of the flat torus and in the case of constant negative curvature. In the general case, this seems to be a difficult question to answer. However, we can give an answer to the related, easier question, whether a smooth function ϕ invariant under all intrinsic parallel transports is necessarily constant.

Proposition 4.2.6. Let M be a non-ﬂat, compact Riemannian manifold with non- positive curvature operator R. Then the following statements hold: 4.2. An invariance property of smooth functions on oriented Grassmannians 39

(ii) If M is not one of the exceptions in i , then, for all k dim M, any smooth p q ď function on Gk M which is invariant under intrinsic parallel transport is nec- orp q essarily constant.

Proof. i Let M be a Kähler manifold of dimension 2n 4. The almost complex p q ě structure J is parallel and it gives rise to a smooth function ϕ on oriented 2-planes, which is invariant under parallel transports but it is not constant. This function is defined via ϕ Aor v ,Jv where v ,v is an oriented orthonormal basis of p q “ x 1 2y 1 2 2 Aor G M . P orp q Now, let M be a Quaternion-Kähler manifold of real dimension 4n 8 with ě non-positive curvature operator. The canonical 4-forms Ω (see for example [Ish74] or [Gray69]) globally defined on M is parallel. This gives rise to the smooth function ϕ :

4 G M R, via ϕ Aor Ωp v ,...,v where v ,...,v is an oriented orthonormal orp q Ñ p q“ p 1 4q 1 4 4 basis of Aor G M and Aor TpM. This function is invariant under parallel P orp q P transports and non constant. Finally, let M be a locally symmetric space of non-constant non-positive curvature, then M is a compact quotient of a symmetric space with non-constant curvature and its Riemannian curvature tensor is parallel. We deﬁne ϕ C8 G2 M P p orp qq as ϕ Aor R v ,v v ,v where v ,v is an oriented orthonormal basis of Aor p q “ x p 1 2q 2 1y 1 2 P G2 M . Now, ϕ is invariant under all parallel transports but it is not constant. orp q ii If M is a n-dimensional manifold which is not one of the exceptions above, p q the holonomy group of M is SO n (see [Ber93] or [Bes87]). Therefore, any smooth p q function on Gk M invariant under all intrinsic parallel transports is also invariant orp q under the non-intrinsic parallel transports by Theorem 4.2.4 and, hence, is constant due to the transitive action of SO n on oriented k-planes in the tangent space. p q Remark 4.2.7. It seems to be an open question whether there exist compact non- 4.2. An invariance property of smooth functions on oriented Grassmannians 40 locally symmetric Quaterion-K¨ahler manifolds with non-positive curvature operator. We refer the reader to [CDL14,CNS13,LeB91,LeB88] for an overview concerning this question. Part II

41 Chapter 5

Introduction

A graph-like manifold is a family of compact, oriented and connected Riemannian manifolds Xε ε made of building blocks according to the structure of metric graph t u ą0 X0. Roughly speaking, a metric graph is a graph where every edge e is associated to a length ℓe. The manifold Xε has the property that it shrinks to the metric graph in the limit ε 0. More precisely, a graph-like manifold is made up of edge Ñ neighbourhoods Xε,e 0,ℓe Yε,e and vertex neighbourhoods Xε,v according to the “r sˆ structure of the underlying metric graph. The parameter ε can be considered as the radius of the edge neighbourhoods. Xε,e and Xε,v are n-dimensional manifolds with boundary whose intersection is a boundary-less n 1 -dimensional manifold Yε,e if p ´ q the edge e emanates from the vertex v. Yε,e is called the transversal manifold (see Section 6.4 for a complete description). Graph-like manifolds are widely used in Mathematics as well as in Physics. In Mathematics, they are used to prove spectral properties of manifolds. The main example is given by Colin de Verdièr [CdV86] who proved that the first non-zero eigenvalue of a compact manifold of dimension n 3 can have arbitrarily large mul- ě tiplicity. Other authors used them to prove the existence of metric with arbitrarily large eigenvalues. Gentile and Pagliara [GP95] proved that any manifolds of dimension n 4 admits a metric such that the first non-zero eigenvalues of the Hodge ě Laplacian acting on differential forms is arbitrarily large. Similar results were also proved in [CM10] and [CG14] where the authors analysed the first non-zero eigenvalue of the Hodge Laplacian on surfaces with boundary and of the rough Laplacian

42 5.1. Aim and main results 43 on diﬀerential forms on manifolds of any dimension, respectively. The key feature of these ‘spectral engineering’ articles is to modify the manifolds with shrinking ‘pieces’ without aﬀecting its topology and carry out the analysis on the new manifold, that can then be considered as a graph-like manifold. In Physics, graph-like manifolds, or more concrete, small neighbourhoods of metric graphs embedded in R3, are used to model electronic and optic nano-structures. The natural question arising is whether the underlying graph is a good approximation for the graph-like manifold. This leads to the study of the asymptotic behaviour of the eigenvalues of the manifold. In this direction, the convergence of the Laplacian on functions on graph-like manifolds, the scalar Laplacian, has been analysed in details, and the convergence of various objects such as resolvents, spectrum, etc. is established in many contexts (see [EP05], [Pos12], [EP13] and references therein).

Related work. Graph-like manifolds can also be considered as collapsing manifold with no curvature control. In this context, the focus is on understanding how well the manifold approaches its limit space. An example is given in [AC95] where the authors considered manifolds with shrinking handles. Similar partial collapsing have also been considered in [AT12] and [AT14], where the authors studied the limit spectrum of the Hodge-Laplacian of the collapsing of one part of a connected sum. Another research line for collapsing manifolds is to study the collapse under curvature bounds. In general, this assumption gives rise to extra structures, we refer the reader to [Jam05], [Lot02], [Lot14] for a detailed overview.

5.1 Aim and main results

The aim of our investigation is to describe the behaviour of the eigenvalues of the p Hodge Laplacian ∆Xε acting on diﬀerential p-forms on a compact n-dimensional graph-like manifold Xε for p 1,...,n 1 as ε 0. “ ´ Ñ The main idea beyond our description is that any p-forms can be orthogonally decomposed into three components: a harmonic one; an exact one; and a co-exact one (see [deR55] or [McG93] and references therein or Section 6.5). Consequently, the p p,ex p,co´ex spectrum of ∆Xε is the union of the spectra of the ∆Xε and of ∆Xε , the Hodge 5.1. Aim and main results 44

Laplacian acting on exact and co-exact forms, respectively, hence it is suﬃcient to describe the eigenvalues of these two operators in the limit ε 0 to obtain a full Ñ description of the asymptotic behaviour of the eigenvalues of the Hodge Laplacian acting on p-forms on Xε. We will denote the j-th eigenvalue, counted with multiplicity, p,ex p,co´ex p “p of ∆ and ∆ by λ¯ Xε and λ Xε , respectively. For short, we will call Xε Xε j p q j p q them exact and co-exact p-form eigenvalues. By Hodge duality and the fact that the exterior derivative d and its adjoint d˚ are isomorphisms between the space of exact and co-exact eigenforms (see also end of Section 6.5), the exact and co-exact eigenvalues are related as follows.

p “p´1 p “n´p λ¯ Xε λ Xε and λ¯ Xε λ Xε j 1. (5.1.1) j p q“ j p q j p q“ j p q @ ě

Hence, the asymptotic behaviour of the exact p-form eigenvalues up to the middle p degree gives the description of all the eigenvalues of ∆Xε . We observe that for a graph-like manifold of dimension two, the spectrum of the Laplacian in all degree forms is entirely determined by its spectrum on functions. Since the convergence of the function eigenvalues on Xε has already been established [Pos12,EP13], we easily conclude the converge of the eigenvalues of the Laplacian on Xε for all degree forms by (5.1.1) (see Section 7.2, in particular Corollary 7.2.3). For higher dimensional graph-like manifolds, we still have convergence for the 1-form eigenvalues, but it is not suﬃcient to describe the form eigenvalues in higher degrees: by Hodge duality we only obtain the convergence of the co-exact n 1 -form eigenvalues. The remaining p ´ q p-form eigenvalues are all divergent under the hypothesis that Xε is transversally trivial (Theorem 7.3.3), i.e., the transversal manifolds Ye have trivial p-th cohomology groups for p 1,...,n 2. This is proved using the natural structure of Xε and the “ ´ McGowan Lemma stated in Proposition 6.6.4. Removing the topological assumption, we have divergence for some of the higher form eigenvalues, namely, for eigenvalues indexed by j N, where N depends on the dimension of the cohomology of Ye for ě all e E (Theorem 7.3.5). P We summarise our results as follows.

Theorem 5.1.1. Let Xε be a compact n-dimensional Riemannian graph-like manifold and let X0 be its associated metric graph. The following statements are true. 5.1. Aim and main results 45

(i) The 0-form eigenvalues, or equivalently the exact 1-form eigenvalues, of Xε converge to the eigenvalues of X0, i.e.,

¯1 λj Xε λj Xε λj X0 j 1. p q“ p q ÝÑεÑ0 p q @ ě

(ii) Let n 3 and 2 p n 1 and that Xε is a transversally trivial manifold. ě ď ď ´ Then, the ﬁrst exact p-form (co-exact p 1 -form) eigenvalues diverge, i.e, p ´ q ¯p “p´1 λ1 Xε λ1 Xε , p q“ p q ÝÑεÑ0 8 and so do all the exact p-form (co-exact p 1 -form) eigenvalues. p ´ q (iii) Let n 3 and 2 p n 1. Then, j-th exact p-form (co-exact p 1 -form) ě ď ď ´ p ´ q eigenvalues diverge, i.e.,

“ ¯p p´1 p´1 λN Xε λN Xε , for N 1 2 dim H Ye , εÑ0 p q“ p q ÝÑ 8 “ ` ePE p q ÿ and so does all the j-th exact and co-exact form eigenvalues (in the right dimension) for j N. ě We also asked ourselves about the relation between spectral gaps in the spectrum of the Laplacian acting on 1-forms on Xε and X . The interval a,b is a spectral gap 0 p q for the Laplacian if it does not belong to its spectrum. Our asymptotic description, in particular Theorem 7.2.2 and Corollary 8.1.1, yields the following.

Corollary 5.1.2. Assume that the graph-like manifold Xε is transversally trivial and suppose that a ,b is a spectral gap for the metric graph X , then there exist aε, bε p 0 0q 0 with aε a and bε b such that aε,bε is a spectral gap for the Hodge Laplacian Ñ 0 Ñ 0 p q ‚ on Xε in all degrees, i.e., σ ∆ aε,bε . p Xε q X p q“H

In all our applications (presented in Chapter 8) we assume aε a 0, hence “ “ the interval 0,bε is a spectral gap (0 is always an eigenvalue). Considering a single p q graph-like manifold with constant volume, we are able to recover a result of Gentile and Pagliara [GP95] on the divergence of the ﬁrst non-zero p-form eigenvalue (see Proposition 8.2.1, Section 8.2). Moreover, we consider families of graph-like manifolds arising from families of either Ramanujan graphs or general graphs. In this setting, we manage to ﬁnd spectral gap properties in relation to volume properties of the manifolds (Section 8.3). 5.2. Overview of the text 46 5.2 Overview of the text

The material is organised as follows. In Chapter 6 we present all the necessary preliminaries to our analysis. We describe metric graphs and graph-like manifolds together with their associate Laplacians. We also review some basic facts about the Hodge-Laplacian acting on diﬀerential forms on a manifold and we state the Mc- Gowan Lemma (Proposition 6.6.4), the key ingredient in our proofs. In Chapter 7 we state the main results, namely we prove convergence for the exact 1- form eigenvalues and divergence for higher dimensional degree form eigenvalues. For completeness, we also discuss the dimension of the space of harmonic forms. Finally, in Chapter 8 we present some applications. We establish the existence of spectral gaps for graph-like manifolds with underlying metric graph having a spectral gap in the spectrum of its Laplacian. Moreover, we construct examples of (family of) manifolds with (upper or lower) bounds on the ﬁrst eigenvalues of the Laplacian acting on functions or forms. Chapter 6

Preliminaries

We here introduce all the basic notions needed in the next chapters and we set the notation. In Section 6.1 we define discrete graphs and introduce their discrete Laplacians. These definitions are the basis to treat metric graphs and the corresponding Lapla- cians, defined in Section 6.2. In Section 6.3 we describe discrete and metric Ra- manujan graphs, which will be used to construct families of manifolds with “special” spectral gap (see Section 8.3). Section 6.4 is dedicated to the definition of graph-like manifolds and to a brief description of their Hodge Laplacian on differential 1-forms. In Sections 6.5 and 6.6 we recall some facts about the Hodge Laplacian acting on forms and its eigenvalues. In particular, we describe how eigenvalues behave under scaling and we present the McGowan Lemma, an eigenvalue estimate from below. This lemma will be crucial in the proof of the divergence behaviour of the 1-form eigenvalues (see Section 7.3).

6.1 Graphs and their Laplacians

A finite discrete graph is a triple G V,E, where V V G and E E G are “ p Bq “ p q “ p q finite sets, called vertices and edges sets respectively, and : E V V is such that B Ñ ˆ e e, e associates to an edge its initial and terminal vertex. This map fixes ÞÑ pB´ B` q an orientation for the graph, crucial when working with 1-forms. In what follows, we will assume, without stating it each time, that all discrete graphs are connected.

47 6.1. Graphs and their Laplacians 48

For a vertex v V we denote by P E˘ e E e v v “t P | B˘ “ u the set of incoming and outgoing edges at a vertex v, and with

´ ` Ev E E “ v Y¨ v (disjoint union) the set of edges emanating from v. The degree of a vertex is the number of emanating edges, i.e.,

deg v : Ev . “ | | In our graphs, we allow loops, i.e., edges e such that e e v, and each B´ “ B` “ loop is counted twice in deg v and it appears twice in Ev, due to the disjoint union. We also allow multiple edges, i.e., edges e e with the same starting and ending 1 ‰ 2 point. To consider various types of discrete Laplacian, it is convenient to introduce edge and vertex weights, deﬁned as follows.

µ µE : E 0, , e µe 0, “ ÝÑ p `8q ÞÑ ą

µ µV : V 0, , v µv 0. “ ÝÑ p `8q ÞÑ ą The graph G,µ is called weighted discrete graph. There are two natural choices p q for the weights, the combinatorial weight and the standard weight. The combinatorial weight is deﬁned such that µe 1 and µv 1, while the standard weight is such that “ “ µe 1 and µv deg v. “ “ Throughout this dissertation, we will consider the following weights.

´1 ´1 µE ℓ : E 0, , e ℓ 0, “ ÝÑ p `8q ÞÑ e ą

µV deg : V 0, , v deg v. “ ÝÑ p `8q ÞÑ

In particular, we deﬁne the function ℓ : E 0, such that e ℓe as one ÝÑ p `8q ÞÑ over weight and we call it the length function associated to G. Given a function F : V C on the vertex space of G , the discrete Laplacian ÝÑ on functions is deﬁned as 1 1 ∆GF v F ve F v , p qp q“´deg v ℓe p q´ p q ePEv ÿ ` ˘ 6.2.MetricGraphsandtheirLaplacians 49

where ve is the vertex on the opposite side of v on e Ev. P ˚ We observe that ∆G can also be deﬁned as ∆G d dG where “ G

´1 dG : ℓ V, deg ℓ E,ℓ , such that dGF e F e F e , 2p q ÝÑ 2p q p q “ pB` q´ pB´ q and where ℓ V, deg CV and ℓ E,ℓ´1 CE carry the norms given by 2p q“ 2p q“

2 2 2 2 1 F F v deg v and η ´1 η , ℓ2pV,degq ℓ2pE,ℓ q ℓ } } “ vPV | p q| } } “ ePE | | e ÿ ÿ ˚ respectively, and where dG is its adjoint operator with respect to the corresponding inner products.

1 ˚ ´1 We can equally deﬁne a Laplacian on 1-forms by ∆ : dGd , acting on ℓ E,ℓ . G “ G 2p q Remark 6.1.1. For a general weighted graph G,µ , the weighted discrete Laplacian p q is deﬁned as 1 ∆ F v µ F v F v , pG,µq µ e e p q“ v ePE p p q´ p qq ÿv 2 2 acting on the space ℓ2 V,µ F F v vPV F : |F v | µv . p q“t “ p p qq | } } ℓ2pV,µq “ vPV p q ă 8u ř For further readings on discrete graphs and discrete Laplacians we refer the reader to [Pos12, Section 2.1]

6.2 Metric Graphs and their Laplacians

Let G V,E, be a discrete graph and let ℓ : E 0, be the associated length “ p Bq Ñ p 8q function introduced in the previous section.

A metric graph X0 associated to the discrete graph G is the quotient

∼ X0 : ¨ Ie , “ ePE { ď where Ie : 0,ℓe and ∼ is the relation identifying the end points of the intervals Ie “r s according to the graph, i.e., x y if and only if ψ x ψ y where „ p q“ p q

0 Ie e, P ÞÑ B´ $ ψ : ¨ Ie V, ℓe Ie `e, Ñ ’ P ÞÑ B ePE &’ ď x 0,ℓe x. P ¨ ePEp q ÞÑ ’ %’ Ť 6.2.MetricGraphsandtheirLaplacians 50

An alternative way to describe a metric graph is to think of G as a topological graph where each edge e E (topologically an interval) is associated to a length P ℓe 0. ą Metric graphs carry a natural measure on each interval, the Lebesgue measure dse. This allows us to deﬁne a natural Hilbert space of functions and 1-forms by

2 2 2 1 2 1 L X0 L Ie and L Λ X0 L Λ Ie p q“ ePE p q p p qq “ ePE p p qq à à with norms given by

2 2 2 2 2 2 2 1 2 1 f L pX0q : fe L pIeq and α L pΛ pX0qq : αe L pΛ pIeqq } } “ ePE } } } } “ ePE } } ÿ ÿ for functions f : X C, f fe e E and 1-forms α αe e E ge dse e E on 0 ÝÑ “ p q P “ p q P “ p q P X0, respectively. We remark that functions on Ie can be identified with 1-forms via fe fe dse, the difference of forms and functions appears only in the domain of the ÞÑ corresponding differential operators below.

We deﬁne the exterior derivative d dX on X as the operator “ 0 0

2 1 1 d: dom d L Λ X such that d fe e E f dse e E ÝÑ p p 0qq p q P “ p e q P whose domain is dom d H1 X C X , “ p 0qX p 0q 1 2 1 1 2 1 where H X f L X f f e E L X H Ie and C X p 0q“t P p 0q | “ p eq P P p 0qu “ ePE p q p 0q denotes the space of continuous functions on X0. À

It is not diﬃcult to see that d dX is a closed operator with adjoint given by “ 0

˚ 1 d αe e E α e E, p q P “ ´p eq P with domain ˚ 1 1 ñ dom d α H Λ X0 αe v 0 , “ P p p qq ePE p q“ ! ˇ ÿ ) 1 1 2 ˇ 1 1 2 ñ where H Λ X0 α ge dse ePE L Λˇ X0 ge L Ie and α is the p p qq “ t “ p q P p p qq | P p qu oriented evaluation of α, i.e.,

ñ ge 0 , v ´e αe v ´ p q “ B p q“ $ ’ge ℓe , v e. & p q “ B` %’ 6.2.MetricGraphsandtheirLaplacians 51

The (metric) Laplacians acting on functions and 1-forms deﬁned on X0 are the operators ∆0 d˚d and ∆1 dd˚, X0 “ X0 “ 0 1 respectively. The domains of ∆X0 and ∆X0 are

dom∆0 f dom d df dom d˚ , X0 “t P | P u dom∆1 α dom d˚ d˚α dom d , X0 “t P | P u respectively. Writing the vertex conditions for the Laplacian on functions explicitly, we obtain the conditions

fe 0 , v é ñ 1 fe v : p q “ B is independent of e Ev and f v 0, (6.2.1) p q “ $ P ep q“ ’fe ℓe , v e ePEv & p q “ B` ÿ called standard%’ or Kirchhoff vertex conditions. The first condition can be rephrased as continuity of f on the metric graph, while the second is interpreted as a flux

1 conservation where the 1-form df f e is considered as a vector field (we remind “ p eq the reader that there is one-to-one correspondence between vector fields and 1-forms through the musical isomorphism [GHL90, p.75]). As for the discrete Laplacian, it is possible to define a weighted metric Laplacian. We do not give details of this operator and of the various types of metric Laplacians here as we do not treat them. For further details we refer the reader to [Ku04,Ku05] and references therein.

0 1 The Laplacians ∆X0 and ∆X0 are both self-adjoint and non-negative operators and, since X0 is compact, they have purely discrete spectrum [Pos12, Proposition 2.2.10, and 2.2.14]. Moreover, they fulﬁl a supersymmetry condition in the sense of [Pos09, Sec. 1.2], as explained below. Set H H H , a Hilbert space, then d has supersymmetry if d : dom d H “ 0 ‘ 1 ÝÑ 1 2 2 1 and dom d H . In our case, H L X and H L Λ X , and so d dX Ă 0 0 “ p 0q 1 “ p p 0qq “ 0 has supersymmetry.

0 1 As a consequence, the spectra of ∆X0 and ∆X0 away from zero coincide including multiplicity, i.e., let λ0 X and λ1 X denote the eigenvalues of ∆0 and ∆1 in j p 0q j p 0q X0 X0 6.3. Ramanujan Graphs 52 increasing order and repeated according to their multiplicity, then

λ1 X λ0 X j 1. (6.2.2) j p 0q“ j p 0q @ ě

A general proof of this fact can be found in [Pos09, Proposition 1.2]. We conclude this section presenting a relation between discrete and metric Lapla- cian for equilateral graphs [Nic85,Cat97].

A graph is said to be equilateral if ℓe ℓ for all e E and the spectra of its “ 0 P discrete and metric Laplacian on functions (or 0-forms) satisfy the following. Let Σ : jπ ℓ 2 j 1, 2,... be the Dirichlet spectrum of the interval 0,ℓ , “ tp { 0q | “ u r 0s then

0 λ σ ∆ if and only if φ λ : 1 cos ℓ ?λ σ ∆G (6.2.3) P p X0 q p q “ ´ p 0 q P p q for all λ Σ. R As a consequence of the supersymmetry condition mentioned above, the same relation holds for the spectra of discrete and metric Laplacian on 1-forms.

There is also a relation at the bottom of the spectrum of ∆G and ∆X0 for general (not necessarily equilateral) metric graphs for which we refer to [Pos09, Sec. 6.1] or [Pos12, Sec. 2.4.2] for more details.

6.3 Ramanujan Graphs

A discrete graph G is k-regular, if all its vertices have degree k. For ease of notation we assume here that the graph G V,E, is simple, and we write v w for “ p Bq „ adjacent vertices.

Deﬁnition 6.3.1. Let G be a k-regular discrete graph with n vertices and let ∆G be its (normalised) discrete Laplacian. The graph G is said to be Ramanujan if

2?k 1 max 1 µ µ σ ∆G ´ . | ´ | P p q ď k ˇ ( ˇ We remark that many authors use the eigenvalues of the adjacency matrix AG as the spectrum of a graph. The adjacency matrix is given by AG v,w 1 if v w and p q “ „ AG v,w 0 otherwise. As v w is equivalent with w v, the adjacency matrix is p q “ „ „ 6.3. Ramanujan Graphs 53 symmetric. For more details about the adjacency matrix and its spectra we refer the reader to [BH12]. For a k-regular graph, we have the relation 1 AG k id ∆G , or, ∆G id AG (6.3.4) “ p ´ q “ ´k with the discrete graph Laplacian (with length function ℓe 1). We observe that “ σ AG k, k and σ ∆G 0, 2 . Moreover, k and 0 are always eigenvalues of p q Ă r´ s p qĂr s AG and ∆G, respectively, as well as k and 2 are always eigenvalues of AG and ∆G, ´ respectively, if and only if the graph is bipartite (recall that we assume that G is a finite graph). We define the (maximal) spectral gap length of a discrete graph by 1 γ G : min µ, 2 µ µ σ ∆G 0, 2 1 max α α σ AG , α k , p q “ ´ P p qzt u “ ´ k | | P p q | |ă ˇ ( ˇ (6.3.5)( ˇ ˇ i.e., γ G is the distance of the non-trivial spectrum of the Laplacian ∆G from the p q extremal points 0 and 2. Hence, a graph is Ramanujan if its spectral gap length has size at least 2?k 1 γ G 1 ´ . p qě ´ k It has been shown that the lower bound is optimal, i.e., for any k-regular graph (or even for any graph with maximal degree k) with diameter large enough, the spectral gap length is smaller than 1 2?k 1 k η, where 1 η is of the same order as ´ ´ { ` { the diameter (see [Nil91, Thm. 1] and references therein). In this sense, Ramanujan graphs are optimal expanders, i.e., optimal highly connected sparse graphs. Expander graphs have been characterised in several ways in a number of different contexts and are used in a number of applications in pure mathematics as well as in computer science. For a survey on expander graphs we refer the reader to [HLW06, Lub10, Lub12] and references therein.

i The existence of inﬁnite families G i N of k-regular Ramanujan graphs has been t u P shown whenever k is a prime or a power of a prime (see e.g. [LPS88,Mar88,Mor94]). The existence of inﬁnite families of bipartite k-regular Ramanujan graphs for every k 2 has been proved in [MSS15a] by showing that any bipartite Ramanujan graph ą has a 2-lift which is again Ramanujan, bipartite and has twice as many vertices. Recently, the same authors proved the existence of bipartite Ramanujan graphs of 6.4.Graph-likeManifoldsandtheirHodgeLaplacian 54 every degree and every number of vertices [MSS15b] by showing that a random m- regular bipartite graph, obtained as a union of m random perfect matchings across a bipartition of an even number of vertices, is Ramanujan with nonzero probability.

i Let G i N be a family of Ramanujan graphs such that t u P

i νi : V G (6.3.6) “ | p q| Ñ 8

i and consider the associated family of equilateral metric graphs X i N of length ℓ . t 0u P 0 By (6.2.3), the metric graph Laplacians ∆ i all have a spectral gap X0

h 2 2?k 1 a0,b0 0, 2 with h hk : arccos 1 ´ 0 (6.3.7) p q“ ℓ0 “ “ ´ k ą ´ ¯ ´ ¯ at the bottom of the spectrum.

6.4 Graph-like Manifolds and their Hodge Lapla- cian

A graph-like manifold associated with a metric graph X0 is a family of oriented and connected n-dimensional Riemannian manifolds Xε ε ε (ε small enough) p q0ă ď 0 0 shrinking to X as ε 0 in the following sense. We assume that Xε decomposes as 0 Ñ

Xε Xε,e Xε,v , (6.4.8) “ ePE Y vPV ď ď where Xε,v and Xε,e are called edge and vertex neighbourhood, respectively. More precisely, we assume that Xε,v and Xε,e are closed subsets of Xε such that

Yε,e e Ev Xε,v Xε,e P X “ $ ’ e Ev, &H R ’ with Yε,e a boundaryless smooth connected% Riemannian manifold of dimension n 1 ´ (see Figure 6.1). We will often refer to Yε,e as the transversal manifold. 6.4.Graph-likeManifoldsandtheirHodgeLaplacian 55

ε Xε X0

Figure 6.1: An example of a two dimensional graph-like manifold Xε with associated graph X0.

Furthermore, we assume that the manifolds Xε,v ,ge,v and Yε,e ,hε,e are con- p q p q formally equivalent to the Riemannian manifolds Xv ,gv and Ye ,he , respectively, p q p q with (constant) conformal factor ε2, i.e.,

2 2 gε,v ε gv and hε,e ε he. (6.4.9) “ “

For short, we will write Xε,v ε Xv and Yε,e ε Ye . “ “ Moreover, we assume that Xε,e is isometric to the product Ie ε Ye . Let gε,e ˆ denotes the metric on Xε,e , it satisﬁes

2 2 gε,e ds ε he. (6.4.10) “ `

We often refer to a single manifold Xε as graph-like manifold instead of the family

Xε ε as in the deﬁnition above. p q Throughout this dissertation, we will assume that voln Ye 1 for all e ´1 “ P E, for simplicity. The general case would lead to the weighted vertex condition ñ 1 voln 1 Ye f v 0 instead of (6.2.1) for the metric graph Laplacian (see ePEv p ´ q ep q “ [Pos12,EP09]ř for details).

We call a graph-like manifold Xε transversally trivial if all transversal manifolds p Ye are Moore spaces, i.e., if H Ye 0 for all 1 p n 2 and all e E, p q “ ď ď ´ P where Hp denotes the p-th cohomology group. We observe that a member of a p¨q transversally trivial graph-like manifold Xε is not necessarily homotopy equivalent to the metric graph X0, as the vertex neighbourhoods do not need to be contractible. 6.4.Graph-likeManifoldsandtheirHodgeLaplacian 56

Below, we give an example of how to construct a transversally trivial graph-like manifold.

Example 6.4.1. Let n 2. For each vertex v ﬁx a manifold Xˆv. Remove deg v ě open balls from Xˆv hence the resulting manifold Xv has a boundary consisting of n´1 deg v many components each diﬀeomorphic to a n 1 -sphere S . For e Ev let p ´ q P n´1 Ye S with a metric such that its volume is 1. As (unscaled) edge neighbourhood, “ we choose X ,e : 0,ℓe Ye with the product metric. Then we can construct a graph- 1 “r sˆ like (topological) manifold X with a canonical decomposition as in (6.4.8) (for ε 1) 1 “ by identifying the e-th boundary component of Xv with the corresponding end of the edge neighbourhood X1,e. By a small local change we can assume that the resulting manifold X is smooth. The corresponding family of graph-like manifolds Xε ε is 1 p q ą0 now given as above by choosing the metric accordingly.

Remark 6.4.2. Given a closed (i.e., compact and boundaryless) manifold X and a metric graph X0, it is possible to turn X into a graph-like manifold with underlying metric graph being X0. As an example, we assume X0 to be a metric ﬁnite tree graph, then X turns into a graph-like manifold letting the tree “grow” out of the original manifold. More formally, we construct a graph-like manifold according to a tree graph and leave one cylinder of a leaf (a vertex of degree 1) “uncapped”. Then, we glue the original manifold X with one disc removed together with the free cylinder. Obviously, the resulting manifold is homeomorphic to the original manifold X and can be turned into the a graph-like manifold by the above choice of metric.

p We can now deﬁne on Xε the corresponding Hodge Laplacian ∆ dδ δd acting Xε “ ` on diﬀerential p-forms. The operators d and δ are the classical exterior derivative and its formal adjoint on manifolds, as unbounded operators in the corresponding L2 spaces. We give further details on the Hodge Laplacian on manifolds in the next section. 6.5. Hodge Theory 57 6.5 Hodge Theory

Let M,g be a compact, oriented and connected n-dimensional Riemannian mani- p q fold. The Riemannian metric g induces the L2-space of p-forms

2 p C 2 2 L Λ M,g ω : M ω L2pΛppM,gq ω g dvolg M p p qq “ ÝÑ } } “ M | | ă 8 ! ˇ ż ) ˇ where ˇ

2 2 ω 2 p ω, ω 2 p : ω dvolg M ω ω, } }L pΛ pM,gqq “ x yL pΛ pM,gqq “ | |g “ ^ ˚ żM żM and denotes the Hodge star operator (depending on the metric g). ˚ The Laplacian on p-forms on M is formally deﬁned as

∆p ∆p dδ δd, pM,gq “ “ ` where d is the classical exterior derivative and δ 1 np`n`1 d is its formal adjoint “ p´ q ˚ ˚ with respect to the inner product induced by g. If M has no boundary, then δ is the L2-adjoint of d and ∆p is a non-negative self- adjoint operator with discrete spectrum denoted by λp M,g (repeated according to j p q multiplicity). We allow the manifold M to have a boundary M, itself a smooth manifold of B dimension n 1. As in the function case, it is possible to impose boundary conditions ´ for functions in the domain of the Hodge Laplacian, called absolute and relative boundary conditions. To do so, we ﬁrst decompose a p-form ω in its tangential and normal components on M, i.e., ω ω ω where ω can be considered as a B “ tan ` norm tan form on M while ω dr ωK with ωK being a form on M and r being the B norm “ ^ B distance from M. B Absolute boundary conditions require that ω satisﬁes

ω 0 and dω 0 norm “ p qnorm “ while relative boundary conditions require

ω 0 and δω 0. tan “ p qtan “ These boundary conditions give rise to two unbounded and self-adjoint operators ∆abs and ∆rel with discrete spectrum, the Hodge Laplacians with absolute and relative 6.5. Hodge Theory 58 boundary conditions, respectively (see e.g. [Cha84] or [McG93]). We remark that for functions, the absolute correspond to Neumann while the relative correspond to Dirichlet boundary conditions. Furthermore, since the Hodge star operator exchanges absolute and relative boundary conditions, there is a correspondence between the spectrum of ∆abs and the spectrum of ∆rel, which allows us to study just one of them to cover both cases. In the sequel, we will only consider absolute boundary conditions if the manifold has a boundary, and hence we will mostly suppress the label abs for ease of notation. p¨q 2 In an L -framework, we consider d and δ0 as unbounded operators, deﬁned as the closures d and δ0 of d and δ0 on

dom d ω C8 Λp M,g dω L2 Λp`1 M,g , “ P p p qq P p p qq dom δ ω C8 Λp M,g ˇ δω L2 Λp´1 M,g ,( ω 0 , 0 “ P p p qq ˇ P p p qq norm “ ˇ ( respectively. The Hodge Laplacian withˇ absolute boundary condition is then given by ∆ ∆abs d δ δ d. “ “ 0 ` 0 For this operator, Hodge Theory is still valid. In particular, the de Rham theorem holds (see [deR55] or [McG93, Sec. 2.1] and references therein), i.e.,

Hp M,g Hp M , p q– p q where Hp M,g is the space of harmonic p-forms (with absolute boundary conditions p q if the boundary is non-empty) and Hp M is the p-th de Rham cohomology, and p q any p-form ω L2 Λp M,g can be orthogonally decomposed into an exact (dω¯), P p p qq “ co-exact (δω) and harmonic (ω0) component, i.e.,

ω dω¯ δω“ ω , (6.5.11) “ ` ` 0 whereω ¯ dom d is a p 1 -form, ω“ dom δ is a p 1 -form and ω is a harmonic p- P p ´ q P 0 p ` q 0 form. Moreover, the Hodge Laplacian leaves these spaces invariant and maps p-forms into p-forms. In particular, we can consider the Hodge Laplacian acting on exact and co-exact p-forms as the operators d δ0 and δ0d, respectively. We call their respective eigenvalues exact and co-exact (absolute) p-form eigenvalues, and we denote them by “ λ¯p M,g and λp M,g , respectively. j p q j p q 6.6.Someusefulfactsabouteigenvalues 59

“ Let E¯p λ ker dδ λ and Ep λ ker δ d λ denote the eigenspaces of p q “ p 0 ´ q p q “ p 0 ´ q exact and co-exact p-forms with eigenvalue λ (as the eigenforms are smooth by elliptic “ regularity, we can omit the closures). Since d is an isomorphism between Ep´1 λ p q and E¯p λ , we have p q p “p´1 λ¯ Xε λ Xε , j 1. (6.5.12) j p q“ j p q @ ě In addition, due to Hodge duality, i.e., the star operator interchanges absolute and relative boundary conditions, the following relation holds

p “n´p λ¯ Xε λ Xε j 1. (6.5.13) j p q“ j p q @ ě

6.6 Some useful facts about eigenvalues

We ﬁnally collect some useful facts about the eigenvalues of the Hodge Laplacian on a Riemannian manifold. In particular, we describe their behaviour under scaling of the metric, their characterization, and we present a crucial estimate from below.

Scaling behaviour

We consider a manifold Mε,gε conformally equivalent to M,g with conformal p q p q 2 2 factor ε (meaning that gε ε g). Again for short, we write Mε εM (see also “ “ Section 6.4). We have the following result for L2 norms of p-forms on M and εM and for the eigenvalues of the Hodge Laplacian on M and εM.

Lemma 6.6.1. Let ω be a p-form on a n-dimensional Riemannian manifold M with metric g, and let εM be the Riemannian manifold M,ε2g , then we have p q 2 n´2p 2 ω 2 p ε ω 2 p and (6.6.14a) } }L pΛ pεMqq “ } }L pΛ pMqq λ¯p εM ε´2λ¯p M . (6.6.14b) j p q“ 1p q 2 ´2p 2 Proof. The ﬁrst assertion follows from the fact that we have w 2 ε w and | |ε g “ | |g n dvol 2 M ε dvolg M pointwise. The second follows from the variational character- ε g “ isation of the j-th eigenvalue of Proposition 6.6.2, as we have the scaling behaviour 2 n´2p 2 2 η 2 p ε η 2 p η 2 p } }L pΛ pεMqq } }L pΛ pMqq ε´2 } }L pΛ pMqq . 2 n´2pp´1q 2 2 θ 2 p´1 “ ε θ 2 p´1 “ θ 2 p´1 } }L pΛ pεMqq } }L pΛ pMqq } }L pΛ pMqq 6.6.Someusefulfactsabouteigenvalues 60

Note that the condition η dθ is independent of the metric, see Proposition 6.6.2. “

Eigenvalue characterisation

Here we present a useful characterisation of eigenvalues of the Hodge Laplacian acting on p-forms due to Dodziuk [Dod82, Prop. 3.1], whose proof can be found in [McG93, Prop. 2.1]. Its advantage is that it does not make use of the adjoint δ of the exterior derivative, and hence no derivation of the metric g or of its coeﬃcients are needed. The metric g enters only via the L2 norms. We remind the reader that a form ω satisﬁes absolute boundary conditions when wnorm 0 and dω norm 0 on the boundary (see also p.57, Section 6.5). “ p q “ Proposition 6.6.2. Let M be a compact Riemannian manifold, then the spectrum of the Laplacian 0 λ¯p λ¯p ... on exact p-forms on M satisfying absolute boundary ă 1 ď 2 ď conditions can be computed by

η,η 2 p ¯p L pΛ pMqq λj M inf sup x y η Vj 0 such that η dθ , p q“ Vj θ,θ L2pΛp´1pMqq P zt u “ !x y ˇ ) ˇ where Vj ranges over all j-dimensional subspacesˇ of smooth exact p-forms and θ is a smooth p 1 -form. p ´ q As a consequence we have (see [Dod82, Prop. 3.3] or [McG93, Lem. 2.2]),

Proposition 6.6.3. Assume that g and g are two Riemannian metrics on M such

2 2 that c´g g c`g for some constants 0 c´ c` , i.e., ď ď ăr ď ă 8 2 2 ˚ c gx ξ,ξ gx ξ,ξ c gx ξ,ξ for all ξ T M and x M, r´ p qď p qď ` p q P x P then the eigenvalues ofr exact p-forms ω with absolute boundary conditions fulﬁl n`2p n`2p 1 c´ ¯p ¯p 1 c` ¯p 2 λj M,g λj M, g 2 λj M,g c´ c` p qď p qď c` c´ p q ´ ¯ ´ ¯ for all j 1. ě r As a result, the eigenvalues λ¯p M,g depend continuously on g in the sup-norm j p q deﬁned, e.g., in [Pos12, Sec. 5.2]. In particular, this proposition allows us to consider also perturbation of graph-like manifolds. For a discussion of possible cases we refer to [Pos12, Sec. 5.2–5.6]). 6.6.Someusefulfactsabouteigenvalues 61

An estimate from below for exact eigenvalues

Finally, we introduce a simpliﬁed but useful version of an estimate from below on the ﬁrst eigenvalue of the exact p-form Laplacian on a manifold by McGowan ( [McG93, Lemma 2.3]) also used by Gentile and Pagliara in [GP95, Lemma 1]. Let M,g be a n-dimensional compact Riemannian manifold without boundary p q m and let Ui be an open cover of M such that Uij Ui Uj have a smooth t ui“1 “ X boundary. Moreover, we denote by

Ii : j 1,...,i 1,i 1,...,m Ui Uj “t Pt ´ ` u | X ‰ Hu the index set of neighbours of Ui. We say that the cover Ui i has no intersection of t u degree r if and only if Ui Ui for any r-tuple i ,...,ir with 1 i 1 X¨¨¨X r “ H p 1 q ď 1 ă m i ir m. We choose a ﬁxed partition of unity ρj subordinate to the 2 ă¨¨¨ă ď t uj“1 open cover and we set dρ : maxi sup dρi x g. } }8 “ xPUi | p q| Furthermore, we denote by λ¯p,abs U the ﬁrst positive eigenvalue on exact p-forms 1 p q p on U satisfying absolute boundary conditions on U. Finally, denote by H Uij the B p q p-th cohomology group of Uij.

m Proposition 6.6.4. Let M and Ui be as above and let p 2. Assume that the t ui“1 ě p´1 open cover has no intersection of degree higher than 2 and H Uij 0 for all i, j. p q“ Then, the ﬁrst positive eigenvalue of the Laplacian acting on exact p-forms (without boundary conditions) on M satisﬁes

´3 ¯p 2 λ1 M m 2 p qě 1 cn,p dρ 1 1 } }8 1 ¯p,abs ¯p´1,abs ¯p,abs ¯p,abs ˜λ1 Ui ` λ1 Uij ` λ1 Ui ` λ1 Uj ¸ i“1 p q jPIi ˆ p q ˙ˆ p q p q˙ ÿ ÿ (6.6.15) where cn,p is a combinatorial constant depending only on p and n.

We remark that these assumptions impose a topological restriction on the manifold as such an open cover does not necessarily exist. Actually, the following general version holds for higher exact eigenvalues.

Proposition 6.6.5. Let M and Ui i be as above and let p 2. Assume that the open t u ě p´1 cover has no intersection of degree higher than 2. We set N dim H Uij 1 “ i,j p q ř 6.6.Someusefulfactsabouteigenvalues 62 and N N 1. Then, the N-th eigenvalue of the Laplacian on exact p-forms on M “ 1 ` satisﬁes

1 λ¯p M N m p p qě 1 dρ 8 1 1 } } 1 ¯p,abs ¯p´1,abs ¯p,abs ¯p,abs i“1 ˜λ1 Ui ` jPI λ1 Uij ` λ1 Ui ` λ1 Uj ¸ ÿ p q ÿi ˆ p q ˙ˆ p q p q˙ The proof of this proposition uses the same argument of the proof of McGowan’s lemma (Lemma 2.3 in [McG93]). The ﬁrst step is to consider λ¯p M as characterised N p q in Proposition 6.6.2 and observe that λ¯p M pη,ηq for η one of its eigenform and N p q ě pθ,θq θ a p 1 -form such that θ dη. The second and main step is to construct θ p ´ q “ such that its L2-norm can be bounded from above. The construction is local and uses the knowledge of eigenvalues on the pieces Ui and on the double intersections

Uij extracted from the Cech-de Rham sequence [BT82, Chapter 2], a generalised Meyer-Vietoris sequence. The argument is then completed using a partition of unity. The generalisation to p-forms is trivial since we have particular assumptions on the cover, i.e., no intersections of degree higher than 2 (see the remark after Lemma 2.3 in [McG93]). Chapter 7

Asymptotic behaviour

We now present the main result of Part II, namely, the asymptotic behaviour of the (non-trivial) spectrum of the Hodge Laplacian of a graph-like manifold. To obtain a full description, it is suﬃcient to analyse the spectrum of the Laplacian acting on exact (resp. co-exact forms) away from zero, due to the orthogonal splitting in (6.5.11). We remark that the dimension of the class of harmonic forms depends on the topological properties of the manifold, and this is the reason why we always consider the non-trivial spectrum. This chapter is organised as follows. In Section 7.1 we describe the space of harmonic forms on a graph-like manifold. In Section 7.2 we review the convergence result for the spectrum of the scalar Laplacian, i.e., the Laplacian acting on functions, from which we will recover a convergence result for the Hodge Laplace spectrum for exact 1-forms. We will then focus on the spectrum of the Hodge Laplacian acting on co-exact 1-forms (see Section 7.3), which is divergent in the limit. To show this behaviour, we will make use of Proposition 6.6.4, assuming the cohomology of the transversal manifolds Ye to be non-trivial, together with asymptotic estimates on the building blocks. The same proposition allows us to study the asymptotic behaviour of the spectrum of the Hodge Laplacian on p-forms for 2 p n 2 under the same ď ď ´ assumption. Finally, we will brieﬂy explain how the same argument works, when no assumptions on Ye are made, using a result of McGowan (see Proposition 6.6.5).

63 7.1. Harmonic forms 64 7.1 Harmonic forms

We ﬁrst analyse the dimension of the class of harmonic forms, eigenforms for the zero (trivial) eigenvalue, for graph-like manifolds. We have already explained in the introduction that this dimension depends on topological properties of the manifold. In particular, the de Rham Theorem (Section 6.5) establishes an isomorphism between the class of harmonic p-forms on Xε and the p-th cohomology group of Xε. Therefore, it is suﬃcient to calculate the cohomology groups of the graph-like manifold, to know the dimension of the class of its harmonic forms in all degrees.

Since the graph-like manifold Xε arises from X0, it is intuitive that Xε will inherit some topological properties of the graph. We will see that for a general graph-like manifold, the dimension of its first cohomology group is the sum of the first Betti number of the graph (also equal to the dimension of its first cohomology gruop) and of the dimension of a subset of the first cohomology group of vPV Xε,v . p For transversally trivial graph-like manifolds, i.e., H Ye Ť 0 for 1 p n 2 p q“ ď ď ´ and for all e E, the following lemma holds. P

Lemma 7.1.1. Let Xε be a transversally trivial graph-like manifold of dimension n with underlying metric graph X0. Then, the cohomology groups of Xε are given by

R k 0,n $ Pt u Hk 1 1 Xε ’ H Xv H X0 k 1,n 1 p q“ ’ vPV p q p q Pt ´ u &’ À k À vPV H Xv k 2,...,n 2 . ’ p q Pt ´ u ’ ’À Proof. We use the natural% decomposition of Xε in (6.4.8) and the Mayer-Vietoris sequence. Set A Xe Ie Ye and B Xv and ﬁx ε 1. Then, “ ePE – ePE ˆ “ vPV “ our graph-like (topological)Ť manifoldŤ is X1. Note thatŤ we have avoided any reference to the metric carried by each space as they do not enter into the topological argument. The Mayer-Vietoris sequence in dimension k is

... Hk X Hk A Hk B Hk A B ... ÝÑ p 1q ÝÑ p q‘ p q ÝÑ p X q ÝÑ

k k We have H A H Ye and, by the assumption on the transversal man- p q– ePE p q ifolds, Hk A 0 for kÀ 1,...,n 2, and H0 A Hn´1 A R|E|. p q“ “ ´ p q“ p q– 7.2.Convergenceforfunctionsandexact1-forms 65

k k 0 |V | We also have H B H Xv . In particular, H B R . p q– vPV p q p q– k 2|E| Finally, A B À Ye . Hence, H A B R for k 0,n 1 , X – vPV ePEv p X q – P t ´ u and Hk A B 0 otherwise.Ť Ť p X q“ By compactness, we derive H0 X R. Then, since the long exact sequence p 1q – k k splits into short exact sequences, we obtain H X H Xv for 2 k n 1. p 1q– vPV p q ď ď ´ 1 1 1 A dimensional argument yields H X1 H XÀv H X0 . p q– vPV p q p q The use of Poincar´eduality concludes theÀ claim. À

We observe that this computation agrees with the results in [AC95] where the authors considered a manifold with shrinking handles, i.e., a graph-like manifold where the shrinking parameter involves only the edge neighbourhood. We also remark that, although the dimension of the class of harmonic 0-forms on

0 Xε coincides with the dimension of the ones on the graphs (H X Z since the p 0q “ graph is connected), for harmonic forms of higher degree this is not valid. The class of the harmonic p-forms on Xε is larger than the graph’s one, both in the transversal trivially case and in the general case.

When some or all of the Ye have non-trivial p-th cohomology groups for 1 p ď ď n 2, we do not have a general formula. In this case, the Mayer-Vietoris sequence ´ and a dimensional argument only allows us to deduce the dimension of the ﬁrst cohomology group of Xε, equal to b X q where b X |E| |V | 1 is the ﬁrst 1p 0q` 1p 0q“ ´ ` 1 Betti number of X and q is the dimension of a subset of H Xv . However, 0 vPV p q it is possible to compute the cohomology groups explicitlyÀ for concrete examples of edge and vertex neighbourhoods.

7.2 Convergence for functions and exact 1-forms

The Laplacian on functions on graph-like manifolds has been analysed in details in a series of papers [EP05,EP09,EP13,Pos06,Pos12] where the convergence of several objects has been established. For the proof of the eigenvalues convergence we par- ticularly refer to [EP05] (see also [Pos12]) where the authors proved the following (based on the results [KuZ01,RS01]). 7.2.Convergenceforfunctionsandexact1-forms 66

Proposition 7.2.1 ( [EP05,Pos12]). Let Xε be a compact graph-like manifold associated with a metric graph X and let λj Xε and λj X denote the eigenvalues (in 0 p q p 0q increasing order, repeated according to their multiplicity) of the Laplacian acting on functions on Xε and on X0. Then we have

1{2 λj Xε λj X O ε ℓ for all j 1, | p q´ p 0q| “ p { 0q ě where ℓ0 min ℓe, 1 0 denotes the truncated minimal edge length. Moreover, the “ ePE t uą error depends only on j, and the building blocks Xv, Ye of the graph-like manifold.

The proof in [EP05] is based on two sides eigenvalue estimates obtained by an av- erage process on the vertex neighbourhoods Xε,v which corresponds to projection onto the lowest (constant) eigenvalue using the variational principle or min-max principle as in Proposition 6.6.2. In following works, the convergence of the resolvents (in a suitable sense), spectral projections, eigenfunctions, and discrete and essential spectrum has been proved using an abstract setting that deals with operators acting in different Hilbert spaces (first used in [Pos06] without requiring compactness of the manifold). The basic idea is that we need to define a “distance” between the operators ∆Xε and ∆X0 with suitable identification operators. For a detailed overview and proofs of these techniques we refer to the reader to [Pos06] and [Pos12, Ch. 4]. We have already noticed in (6.2.2) and (6.5.12), that the exact 1-form eigenvalues equal the 0-form (function) eigenvalues both on the graph X0 and on the graph-like manifold Xε. Therefore, the previous result immediately gives the convergence for exact 1-forms, using a simple supersymmetry argument as in [Pos09, Sec. 1.2].

Theorem 7.2.2. Let Xε be a graph-like manifold with underlying metric graph X0. 1 1 Denote by λ¯ Xε and λ¯ X the j-th exact 1-form eigenvalue on Xε and X , respec- j p q j p 0q 0 tively. Then, ¯1 ¯1 λj Xε λj X0 for all j 1. p q ÝÑεÑ0 p q ě Proof. We will just show that the eigenspaces for non-zero eigenvalues of ∆1 ∆1 Xε “ “ dd˚ and ∆0 ∆0 d˚d are isomorphic (the argument works for ε 0 and ε 0 as Xε “ “ ą “ well). Then, the convergence result follows immediately from Proposition 7.2.1. 7.3. Divergence for co-exact p-forms 67

As isomorphism, we choose

d: ker ∆0 λ ker ∆1 λ p ´ q ÝÑ p ´ q for λ 0. First, note that if f ker ∆0 λ , then ‰ P p ´ q ∆1df dd˚df d∆0f λdf, “ “ “ i.e., df ker ∆1 λ , hence the above map is properly deﬁned. The map d as above P p ´ q is injective. If df 0 for f ker ∆0 λ then λf ∆0f d˚df 0. As λ 0 we “ P p ´ q “ “ “ ‰ have f 0. For the surjectivity, let α ker ∆1 λ . Set f : λ´1d˚α (we use again “ P p ´ q “ that λ 0). Then, ‰ df d λ´1d˚α λ´1∆1α α, “ p q“ “ i.e., d as above is surjective. In particular, we have shown that the spectrum of ∆0 and ∆1 away from 0 is the same, including multiplicity.

We remark that if n dim Xε 2, the above theorem is suﬃcient to determine the “ “ spectra of Laplacian in all degree forms. In fact, by (6.5.12) and (6.5.13), the exact 1- form eigenvalues coincide with the 0-form eigenvalues, the co-exact 1-form eigenvalues coincide with the (exact) 2-form eigenvalues, and these eigenvalues coincide with the 0-form eigenvalues. Therefore, we can state the following.

Corollary 7.2.3. Let Xε be a graph-like Riemannian compact manifold of dimension

2 associated to a metric graph X0. Then,

¯1 λj Xε λj X0 , p q ÝÑεÑ0 p q (7.2.1) “ 1 2 λj Xε λj Xε λj Xε λj X0 , p q“ p q“ p q ÝÑεÑ0 p q for all j 1. ě In addition, by Hodge duality (see (6.5.13)), Theorem 7.2.2 gives convergence for n-forms on graph-like manifolds of any dimension.

7.3 Divergence for co-exact p-forms

If n 3, the behaviour of the co-exact p-forms for 1 p n 2 cannot be known ě ď ď ´ using duality. In order to study their limit behaviour, we analyse the limit behaviour 7.3. Divergence for co-exact p-forms 68 of the exact p 1 -forms eigenvalues, due to (6.5.12). In particular, we ﬁrst give p ` q some eigenvalue asymptotics for eigenvalues of exact p-forms with absolute boundary conditions on the building blocks of the graph-like manifold, which are needed to make use of Proposition 6.6.4 and Proposition 6.6.5.

7.3.1 Eigenvalue asymptotics on the building blocks

A vertex neighbourhood Xε,v is conformally equivalent to Xv by deﬁnition. As a result of Lemma 6.6.1, we have the following corollary.

Corollary 7.3.1. Let Xε,v be a vertex neighbourhood of a graph-like manifold Xε. Then, the smallest positive eigenvalue of the Laplacian acting on exact p-forms on

Xε,v with absolute boundary conditions satisﬁes

p ´2 p λ¯ Xε,v ε λ¯ Xv . (7.3.2) 1p q“ 1p q

To describe the asymptotic behaviour of the edge neighbourhood, there is a bit more work to do. We recall that the edge neighbourhood Xε,e is isomorphic to Ie Yε,e ˆ with the product metric. However, we cannot make use of the product structure as it does not respect exact and co-exact forms.

Proposition 7.3.2. Let Xε,e be an edge neighbourhood of a n-dimensional graph-like manifold Xε. Then, the smallest eigenvalue of the Laplacian acting on exact p-forms (2 p n 1) with absolute boundary conditions satisﬁes ď ď ´

p ´2 λ¯ Xε,e ε cp ε , (7.3.3) 1p q“ p q

p p where cp ε λ¯ Ye 0 as ε 0, and where λ¯ Ye denotes the ﬁrst eigenvalue p q Ñ 1p q ą Ñ 1p q of the Laplacian acting on exact p-forms on Ye .

Proof. By Proposition 6.6.2 we have to analyse the quotient η 2 θ 2 for an exact } } {} } p-form η and a p 1 -form θ such that η dθ. Recall that Xε,e Ie ε Ye (i.e., p ´ q “ “ ˆ 2 2 Ie Ye with metric gε,e ds ε he). Then, the p 1 -form θ on Xε,e can be written ˆ “ ` p ´ q uniquely as θ θ ds θ (7.3.4) “ 1 ^ ` 2 7.3. Divergence for co-exact p-forms 69

where θ and θ are a p 2 -form and p 1 -form on Ye , respectively. Using the 1 2 p ´ q p ´ q scaling behaviour of the metric in a similar way as in Lemma 6.6.1, we have

2 2 θ 2 p´1 θ dvol Xε,e } }L pΛ pXε,e qq “ | |gε,e żXε,e ´2pp´2q 2 ´2pp´1q 2 n´1 ε θ ε θ ε ds dvol Ye (7.3.5) “ | 1|he ` | 2|he żIe żYe n´2p`1` 2 2 2 ˘ ε ε θ θ ds dvol Ye , “ | 1|he ` | 2|he żIe żYe ` ˘ 2 where the ε factor appears due to the scaled metric ε he. The decomposition of dθ according to (7.3.4) is given by

dθ dYe θ sθ ds dYe θ . (7.3.6) “ p 1 ` B 2q^ ` 2 Hence,

2 2 dθ 2 p dθ dvol Xε,e } }L pΛ pXε,e qq “ | |gε,e żXε,e ´2pp`1q 2 ´2p 2 n´1 ε dYe θ sθ ε dYe θ ε ds dvol Ye “ | 1 ` B 2|he ` | 2|he żIe żYe n´2p´1` 2 2 2 ˘ ε ε dYe θ sθ dYe θ ds dvol Ye . “ | 1 ` B 2|he ` | 2|he żIe żYe ` ˘ (7.3.7)

In particular, if we substitute (7.3.5) and (7.3.7) into the quotient η 2 θ 2 we } } {} } conclude

2 2 2 2 dθ 2 p ε dYe θ1 sθ2 dYe θ2 ds dvol Ye } }L pΛ pXε,e qq ε´2 Ie Ye | ` B |he ` | |he . 2 2 2 2 θ 2 p´1 “ ε θ1 θ2 ds dvol Ye } }L pΛ pXε,e qq ş ş ` Ie Ye | |he ` | |he ˘ In particular, together with Propositionş ş ` 6.6.2 this yields˘

p ´2 λ¯ Xε,e ε cp ε 1p q“ p q with

θ θ1 ds θ2 0, 2 2 2 “ ^ ` ‰ ε dYe θ θ dYe θ ds dvol Ye Ie Ye 1 s 2 he 2 he cp ε sup | ` B | ` | | θ1 p 2 -form, . p q“ ε2 θ 2 θ 2 s p ´ q # I Ye 1 he 2 he d dvol Ye ˇ + ş ş `e | | ` | | ˘ ˇ ˇ θ2 p 1 -form ş ş ` ˘ ˇ p ´ q ˇ In the limit ε 0, this constant tends to a number cp 0 given by Ñ p q 2 dYe θ2 ds dvol Ye Ie Ye | |he cp 0 sup 2 θ2 0 p 1 -form . p q“ θ ds dvol Ye ‰ p ´ q # Ie Ye 2 he ˇ + ş ş | | ˇ ˇ ş ş ˇ ˇ 7.3. Divergence for co-exact p-forms 70

This constant is the min-max characterisation of the ﬁrst eigenvalue of the opera-

p 2 2 p tor id ∆¯ acting on L Ie L Λ Ye , whose spectrum agrees with the spectrum b Ye p qb p p qq p p of ∆¯ (see e.g. [RS78, Thm. XIII.34]). Hence, we have cp 0 λ¯ Ye . Ye p q“ 1p q

7.3.2 Main theorems

We are now ready to prove the divergence behaviour of the spectrum of the co-exact p-forms, for 1 p n 2, on a n-dimensional graph-like manifold Xε. For the rest ď ď ´ of this section we assume that n 3, as the 2-dimensional case has been already ě explained in Corollary 7.2.3. We remind the reader that by (6.5.12), we analyse the exact p-forms for 2 p n 1. ď ď ´ We ﬁrst analyse the case when Xε is transversally trivial, making use of Proposi- tion 6.6.4. Let

Uε Uε,v v V Uε,e e E “t u P Yt u P be an open cover of Xε, where Uε,v and Uε,e are open ε-neighbourhoods of Xε,v and

Xε,e in Xε, respectively, or in other words, a slightly enlarged vertex and edge neighbourhoods to ensure that Uε is an open cover.

It is easy to see that Uε has intersections up to degree 2 only (three or more diﬀerent sets of Uε have always trivial intersection). The intersections of degree 2 are given by Xε,v,e Uε,v Uε,e which is empty if e Ev or otherwise isometric to the “ X R product 0,ε Yε,e , hence conformally equivalent to the product 0, 1 Ye with p qˆ p qˆ 2 conformal factor ε , as we enlarged Xε,v by an ε-neighbourhood. Moreover, Xε,v,e is homeomorphic to 0, 1 Ye , and hence homotopy equivalent to Ye . In particular, p qˆ p´1 p´1 H Xε,v,e H Ye . p q“ p q

Theorem 7.3.3. Let Xε be a graph-like manifold of dimension n 3 with underlying ě metric graph X . Assume that 2 p n 1 and that the p 1 -th cohomology group 0 ď ď ´ p ´ q p´1 of the transversal manifold Ye vanishes for all e E, i.e., H Ye 0. Then, the P p q“ ﬁrst eigenvalue of the Hodge Laplacian acting on exact p-forms on Xε satisﬁes

p ´2 λ¯ Xε τpε , 1p qě where τp 0 is a constant depending only on the building blocks Xv and Ye of the ą graph-like manifold, the truncated minimal length ℓ0 min ℓe, 1 , and p. In particu- “ ePE t u 7.3. Divergence for co-exact p-forms 71

p “p´1 lar, all eigenvalues λ¯ Xε of exact p-forms and all eigenvalues λ Xε of co-exact j p q j p q p 1 -forms tend to as ε 0. p ´ q 8 Ñ

Proof. We apply Proposition 6.6.4 as the cover Uε has no intersections of degree p´1 p´1 higher than 2 and H Xε,v,e H Ye 0. p q“ p q“ We ﬁrst look at the denominator of the right hand side of the estimate in Propo- sition 6.6.4. We note that our open cover Uε is labelled by v V and e E and P P therefore, the sum over i 1,...,m of the estimate in Proposition 6.6.4 becomes a “ sum over v V and e E. Moreover, the sum over the edges can easily be rewritten P P as a sum over the vertices taking care of some appearing extra factors. Using these observations and equations (7.3.2) and (7.3.3), we obtain

2 1 cn,p dρε 1 1 } }8 1 λ¯p ¯p´1 λ¯p λ¯p vPV ˜ 1 Xε,v ` ePE λ1 Xε,v,e ` 1 Xε,v ` 1 Xε,e ¸ ÿ p q ÿv ˆ p q ˙ˆ p q p q˙ 2 1 cn,p dρε 1 1 } }8 1 λ¯p ¯p´1 λ¯p λ¯p ` ePE ˜ 1 Xε,e ` v“B e λ1 Xε,v,e ` 1 Xε,v ` 1 Xε,e ¸ ÿ p q ÿ˘ ˆ p q ˙ˆ p q p q˙ 2 1 deg v cn,p dρε 1 1 2 } }8 1 λ¯p λ¯p ¯p´1 λ¯p λ¯p “ vPV ˜ 1 Xε,v ` 1 Xε,e ` ePE λ1 Xε,v,e ` 1 Xε,v ` 1 Xε,e ¸ ÿ p q p q ÿv ˆ p q ˙ˆ p q p q˙ 2 2 2 1 deg v cn,pε dρε 8 1 1 2 ε 2 } } 1 : ε Cp ε , λ¯p c ε ¯p´1 λ¯p c ε “ vPV ˜ 1 Xv ` p ` ePE λ1 Xv,e ` 1 Xv ` p ¸ “ p q ÿ p q p q ÿv ˆ p q ˙ˆ p q p q˙ where the extra term with deg v and the factor 2 are due to the transformation of the sum over the edges into a sum over the vertices.

We now analyse the constant Cp ε as ε 0. p q Ñ p First, we have seen in Proposition 7.3.2 that cp ε λ¯ Ye 0. Moreover, the p q Ñ 1p q ą norm of the derivative of the partition of unit norm depends on ε as these functions have to change from 0 to 1 on a length scale of order ε on the vertex neighourhoods and on a length scale of order ℓ0 on the edge neighourhood, hence the derivative is ´1 ´1 2 2 2 of order ε ℓ and ε dρε O 1 O ε ℓ . In particular, Cp ε Cp 0 ` 0 } }8 “ p q` pp { 0q q p q Ñ p q as ε 0 provided ε ℓ remains bounded, where Cp 0 depends only on some data of Ñ { 0 p q the building blocks. Therefore, by Proposition 6.6.4 we can conclude 2´3 λ¯p , 1 Xε 2 p qě ε Cp ε p q which proves the theorem. 7.3. Divergence for co-exact p-forms 72

We observe that, by duality and supersymmetry (see (6.5.12) and (6.5.13)), The- orem 7.3.3 gives divergence for the spectrum of p-forms for 2 p n 2. ď ď ´ We also point out that Theorem 7.2.2 and Theorem 7.3.3 for p 2 gives a “ complete description of the behaviour of the spectrum on 1-forms, and by duality we also have a description of the spectrum on n 1 -forms. We remark that this p ´ q spectrum is partially convergent (exact eigenvalues) and partially divergent (co-exact eigenvalues). Hence, we can state the following.

Corollary 7.3.4. Let Xε be a graph-like Riemannian compact manifold of dimension n 3 associate to a metric graph X . Assume that all transversal manifolds Ye have ě 0 trivial cohomology for p 1,...,n 2. Then, “ ´ “ n´1 ¯1 0 λj Xε λj Xε λj X0 , p q“ p q ÝÑεÑ0 p q “ ¯n´1 1 λj Xε λj Xε , (7.3.8) p q“ p q ÝÑεÑ0 8 p λj Xε , p q ÝÑεÑ0 8 for all j 1 and 2 p n 2, ě ď ď ´ We remark that the case n 2 has been treated in Corollary 7.2.3. “ Removing the assumption of vanishing cohomology groups of the transversal manifolds, the following theorem holds.

Theorem 7.3.5. Let Xε be a graph-like manifold of dimension n 3 with underlying ě metric graph X0. Then, the N-th eigenvalue of the Laplacian acting on exact p-forms on Xε satisﬁes p ´2 λ¯ Xε τpε , N p qě where τp 0 is as before and where ą r

p´1 p´1 r N 1 dim H Ye 1 2 dim H Ye . “ ` vPV ePE p q“ ` ePE p q ÿ ÿv ÿ Its proof follows the line of the previous one with the diﬀerence that we use

Proposition 6.6.5 to estimate a higher eigenvalue for exact p-forms on Xε.

Remark 7.3.6. We point out that the ﬁrst N 1 eigenvalues of the theorem above ´ are strictly positive since we consider the spectrum away from zero. The theorem 7.3. Divergence for co-exact p-forms 73

p “p states that λ¯ Xε λ Xε are divergent for j N in the limit ε 0. However, j p q“ pj´1qp q ě Ñ it remains an open question how the ﬁrst N 1 eigenvalues behave asymptotically p ´ q as ε 0. Ñ Chapter 8

Manifolds with spectral gap

In this chapter we discuss some applications of the asymptotic behaviours described in Chapter 7. We will state some general facts about the existence of spectral gaps in the spectrum of the Hodge-Laplacian on a graph-like manifold Xε in relation to existing spectral gaps in the spectrum of the Laplacian on its associated metric graph

X0. Moreover, we will construct manifolds and families of manifolds with spectral gaps. In Section 8.1 we define the Hausdorff convergence and we state a weaker version of Corollary 7.3.4 in relation to this definition (see Corollary 8.1.1). Moreover, we give the definition of spectral gap and a general result on how to produce graph-like manifolds with spectral gaps in their spectrum. In Section 8.2 we construct manifolds with constant volume and arbitrarily large form eigenvalues, i.e., manifolds with an arbitrarily large spectral gap in their Hodge-Laplacian on p-forms for 2 p n 1. ď ď ´ In Section 8.3 we construct families of manifolds with spectral gaps arising from families of Ramanujan graphs and of arbitrary graphs.

8.1 Hausdorﬀ convergence of the spectrum and spectral gaps

Let A, B R be two compact sets. The Hausdorﬀ distance of A and B is deﬁned as Ă

d A, B : max sup d a,B , sup d b, A , where d a,B : inf a b . (8.1.1) p q “ t aPA p q bPB p qu p q “ bPB | ´ | 74 8.1. Hausdorﬀ convergence of the spectrum and spectral gaps 75

A sequence An n of compact sets An R converges in Hausdorff distance to A p q Ă 0 if and only if d An,A 0 as n . In particular, d An,A 0 if and only if p q Ñ Ñ 8 p 0q Ñ for all λ A there exists λn An such that λ λn 0 and for all x R A 0 P 0 P | 0 ´ | Ñ P z 0 there exists η 0 such that x η,x η An for n sufficiently large (see ą r ´ ` sX “ H e.g. [Pos12, Proposition A.1.6]). In view of this definition, a weaker result than Corollary 7.3.4 is as follows.

Corollary 8.1.1. Let Xε be a transversally trivial graph-like manifold with associated metric graph X . Then, for all λ 0 we have that σ ∆‚ 0,λ converges in 0 0 ą p Xε qXr 0s Hausdorﬀ distance to σ ∆X 0,λ . p 0 qXr 0s In fact, in a compact interval 0,λ , eventually all divergent eigenvalues from r 0s higher forms leave this interval, and the remaining ones converge. Furthermore, we asked ourselves about the relation between spectral gaps in the spectrum of the Laplacian acting on 1-forms on Xε and X , i.e., about intervals a,b 0 p q not belonging to the spectrum. More precisely, a spectral gap of an operator ∆ 0 ě is a non-empty interval a,b such that p q

σ ∆ a,b . p q X p q“H

As a consequence of the asymptotic description of the spectrum in Theorems 7.2.2, 7.3.3 and in Corollary 8.1.1, we have the following result on spectral gaps (i.e., intervals disjoint with the spectrum).

Corollary 8.1.2. Assume that the graph-like manifold Xε is transversally trivial and suppose that a ,b is a spectral gap for the metric graph X , then there exist aε, bε p 0 0q 0 with aε a and bε b such that aε,bε is a spectral gap for the Hodge Laplacian Ñ 0 Ñ 0 p q ‚ on Xε in all degree forms, i.e., σ ∆ aε,bε . p Xε q X p q“H Examples of manifolds with spectral gaps can be generated in diﬀerent ways. In [Pos03,LP08] the authors constructed (non-compact) abelian covering manifolds having an arbitrary large number of gaps in their essential spectrum of the scalar Laplacian, and in [ACP09], the analysis was extended to the Hodge Laplacian on certain cyclic covering manifolds. 8.2.Manifoldswitharbitrarilylargespectralgap 76

One can construct metric graphs with spectral gaps, and hence graph-like manifolds with spectral gaps, with a technique called graph decoration that works as follows. We consider a finite metric graph X0 and a second finite metric graph X0. For each v V X , let X v be a copy of a finite metric graph X . Fix a vertex P p 0q 0 ˆt u 0 r v of X . Then the graph decoration of X with the graph X is the graph obtained 0 r 0 0 r from X0 by identifying the vertex v of X0 v with v. This decoration opens up a r r ˆt u r gap in the spectrum of the Laplacian on function on X0 as described in [Ku05] and r r therefore in its 1-form Laplacian. Consequently, the associated graph-like manifold has a spectral gap in its 1-form Laplacian, and no spectrum away from 0 for higher forms, as all the form eigenvalues diverge. More examples of manifolds with spectral gap and family of manifolds with a spectral gap are given in the next sections.

8.2 Manifolds with arbitrarily large spectral gap

Let Xε ε be a graph-like manifold constructed from a metric graph X with under- p q ą0 0 lying (discrete) graph V,E, . We assume the graph-like manifold to be transversally p Bq p trivial (i.e., H Ye 0 for all 1 p n 2 and for all e E). p q“ ď ď ´ P For simplicity, we assume that X0 is equilateral, i.e., all edge lengths are given by a number ℓ 0. The result can be easily extended to the case when c ℓ ℓe c ℓ ą ´ ď ď ` for all e E and some constants c 0. P ˘ ą We write

aε bε, aε bε, aε h bε (8.2.2) À Á if

aε const bε, aε const bε, const aε bε const aε (8.2.2’) ď ` ě ´ ´ ď ď ` for all ε 0 small enough and constants const independent of ε. ą ˘ We ﬁrst summarise the asymptotic spectral behaviour of a graph-like manifold

Xε and its dependence on the parameters ε, ℓ, V , and E . In particular, for the | | | | volume, the 0-forms (functions), and the exact p-forms and co-exact p 1 -forms, p ´ q 8.2.Manifoldswitharbitrarilylargespectralgap 77 we have

n n´1 vol Xε h ε V ε ℓ E (8.2.3a) | |` | | 1{2 0 0 ε λj Xε λj X0 ℓ0 min ℓ, 1 (8.2.3b) | p q´ p q| À ℓ0 p “ t uq p “p´1 1 λ¯ Xε λ Xε 2 p n 1, (8.2.3c) 1p q“ 1 p qÁ ε2 E 1 ε2 ℓ2 ď ď ´ | |p ` { q

where the constants in etc. depend only on the building blocks Xv and Ye of the À (unscaled, i.e, ε 1) graph-like manifold. Equation (8.2.3a) is a direct consequence “ of the structure of the graph-like manifold described in (6.4.8). Equation (8.2.3b) is a direct consequence of Proposition 7.2.1. Equation (8.2.3c) follows from analysing the lower bound constant τp in Theorem 7.3.3 (or Theorem 7.3.5). We see that the constant Cp ε in its proof is bounded from above by p q

2 2 2 2 Cp ε V E 1 ε ℓ E 1 ε ℓ , p qÀ | | ` | |p ` { q À | |p ` { q ` ˘ where again the constants in depend only on the building blocks and where we À used V deg v 2 E for any graph G, assuming that there are no isolated | | ď vPV “ | | vertices, i.e.,ř vertices of degree 0. γ We now assume that ℓ ℓε ε depends on ε for some γ R (negative γ’s are “ “ P γ not excluded). In particular, X0 now also depends on ε, and we write ε X0 for a metric graph with all edge lengths multiplied by εγ. If we plug ℓ εγ into equations “ (8.2.3a)–(8.2.3c), we observe the following.

(i) In (8.2.3a), the dominant term is εn for γ 1 and it is εn´1`γ otherwise. ě (ii) For the metric graph eigenvalues, we have λ0 εγX ε´2γλ0 X . j p 0q“ j p 0q (iii) In (8.2.3b) we need γ 1 2 for convergence to hold, as the error term is of ă { order ε1{2 min εγ, 1 ε1{2´maxtγ,0u. We also need γ 1 4 for the metric { t u “ ą ´ { graph eigenvalue (of order ε´2γ) to be dominant with respect to the error (of order ε1{2´maxtγ,0u).

(iv) In (8.2.3c) we need γ 2 for divergence to hold. Moreover, ε2 is the dominant ă term in the denominator of the RHS for γ 1, and it is ε4´2γ otherwise. ď 8.2.Manifoldswitharbitrarilylargespectralgap 78

Therefore, equations (8.2.3a)–(8.2.3c) become

εn´1`γ E , γ 1 n n´1`γ vol Xε h ε V ε E h | | ď (8.2.3a’) | |` | | $ ’εn V , γ 1, & | | ě %’ h ε´2γ, 1 4 γ 1 2 0 λ Xε ´ { ă pă { q (8.2.3b’) j p q $ ’ ε1{2, γ 1 4, &À ď´ { ’ % ε´2, γ p 1 λ¯ Xε ď (8.2.3c’) 1p qÁ $ ’ε´4`2γ, 1 γ 2 . & ď pă q %’ These equations and statements (i)–(iv) give the existence of manifolds with constant volume and arbitrarily large form eigenvalues, i.e, manifolds with an arbitrarily large spectral gap in their form spectrum. Our proposition below states an analogous result than the one in [GP95, Theorem 1], where the authors state that for any closed manifold M of dimension n 4 there exits a metric of volume 1 such that the first ě non-zero p-form eigenvalue λp M is unbounded. In particular, they give an answer to 1p q a question of Tanno [Tan83], whether there exists a constant k M such that the first p q non-zero p-form eigenvalue satisfies λp M k M vol M,g ń{2 for all Riemannian 1p qď p qp p qq metrics g on M. The same question was previously posed by Berger [Ber73] on the first non-zero function eigenvalue and answered positively (see [GP95] and references therein for further contributions). We observe that the construction of Gentile and Pagliara in [GP95] corresponds to a simple graph with one edge and two vertices. Therefore, we conclude the following.

Proposition 8.2.1. On any transversally trivial graph-like manifold of dimension n 3 there exists a family of metrics gε of volume 1 such that for the ﬁrst eigenvalue ě on exact p-forms we have r p λ¯ X, gε as ε 0 1p q Ñ 8 Ñ for 2 p n 1. Moreover, ther function (p 0) and exact 1-form spectrum ď ď ´ “ converges to 0, i.e.,

0 1 λ X, gε λ¯ X, gε 0 as ε 0. 1p q“ 1p q Ñ Ñ r r 8.3. Families of manifolds with special spectral properties arising from families of graphs 79

Proof. Let gε be the metric of the graph-like manifold as constructed in Section 6.4. For any γ 1, we have ă

p 2{n ´2 2pn´1`γq{n ´2p1´γq{n λ¯ X,gε vol X,gε ε ε ε as ε 0 1p qp p qq Á “ Ñ 8 Ñ

´2{n by (8.2.3c’) and (8.2.3a’). Set now gε : vol X,gε gε, then vol X, gε 1 and “ p q p q“

p 2{n p ´2p1´γq{n λ¯ X, gε vol X,gε λ¯r X,gε ε as ε r 0. 1p q“ p q 1p qÁ Ñ 8 Ñ If 1 4 γ r1 2, then the 0-form (and exact 1-form) eigenvalues of the metric ´ { ă ă { 0 ´2γ graph and the manifold are close and λ X,gε h ε , hence j p q

0 2{n 0 2pn´1`γq{n ´2γ 2pn´1qp1´γq{n λ X, gε vol X,gε λ X,gε h ε ε ε 0. j p q“ p q j p q “ Ñ We observer that for manifolds as constructed in the proof, the transversal length

p1´γq{n scale (the one of the transversal manifolds Ye) is ε 0, while the longitudinal Ñ ´p1´1{nqp1´γq length scale (the one of the metric graph edges Ie) is ε as ε 0. Ñ 8 Ñ This implies that the edge neighbourhoods become thinner but longer in the limit. Unfortunately, we cannot extend the result of [GP95] to the case n 3 and “ 1-forms, as the exact 1-form spectrum converges.

8.3 Families of manifolds with special spectral properties arising from families of graphs

We now consider families of graph-like manifolds constructed according to a sequence

i of graphs G i N. We assume for simplicity that the vertex degree is uniformly t u P bounded, say by k N. Then, if there are no isolated vertices, we have 0 P

i i i |V G | deg i v 2|E G | 2k |V G |, p q ď G “ p q ď 0 p q v V Gi Pÿp q i i i.e., νi : |V G | |E G | as i . “ p q » p q Ñ 8 We begin with a general statement about the spectral convergence. We assume

i i that G i N is a family of discrete graphs and that X i N is the family of associated t u P t 0u P i equilateral metric graphs, each graph X0 having edge lengths equal to ℓi (for the deﬁnition of equilateral graph, see Section 6.2). Accordingly, we construct a family 8.3. Families of manifolds with special spectral properties arising from families of graphs 80

i of graph-like manifolds X i N where the building blocks Xv and Ye are isometric t ε u P to a given number of prototypes (independent of i), such that Ye all have trivial cohomology for 1 p n 2 (see Example 6.4.1), so that all graph-like manifolds ď ď ´ i Xε are transversally trivial and hence our estimates (8.2.3a)–(8.2.3c) are uniform in the building blocks and (8.2.3c) holds for the ﬁrst exact eigenvalue. We call such a family of graph-like manifolds uniform.

i We now assume that εi and ℓi are dependent on the number of vertices νi of G . Speciﬁcally, we set ´α ´β εi ν and ℓi ν (8.3.5) “ i “ i for some α 0 and β R (negative values of β are not excluded). In particular, Xi ą P 0 ´β i i now also depends on ε, and we write νi X0 for the metric graph X0 with all edge ´β lengths being νi . Substituting conditions (8.3.5) into equations (8.2.3a)–(8.2.3b), we observe the following.

(i’) The volume is now given by vol Xi h ν´nα`1 ν´pn´1qα´β`1. ε i ` i (ii’) For the metric graph eigenvalue, we have λ0 ν´βXi ν2βλ0 Xi . j p i 0q“ i j p 0q (iii’) In (8.2.3b) we need max β, 0 α 2, for the convergence to hold, as the error t uă { 1{2 ´α{2`maxtβ,0u term is of order ε min ℓi, 1 ν (Figure 8.1 (a) below). We also i { t u“ i need β α 2 and β 0, or β α 4 and β 0, for the metric graph ě ´ { ě ě ´ { ď 2β eigenvalue (of order νi ) to be dominant with respect to the error (of order ´α{2`maxtβ,0u νi ) (Figure 8.1 (b) below).

(iv’) In (8.2.3c), we need α 1 2 (resp. 2α 1 β), for the divergence to hold. ą { ą ` Moreover, if α β the dominant term in the denominator of the RHS is ν´2α`1, ě it is ν´4α`2β`1 otherwise (Figure 8.1 (c) below).

Therefore, in view of the above statements, we can rewrite (8.2.3a)–(8.2.3c) as

ν´pn´1qα´β`1, α β, i h i ě vol Xε $ (8.2.3a”) ’ν´nα`1, α β. & i ď %’ 2β i h ν λj X , β α 2, β 0 or β α 4, β 0 , 0 i i 0 λj Xε p q p ě´ { ě q p ě´ { ď q (8.2.3b”) p q $ ´α{2 ’ ν , otherwise. &À i %’ 8.3. Families of manifolds with special spectral properties arising from families of graphs 81

2α´1 νi , α β, λ¯p Xi ě for 2 p n 1. (8.2.3c”) 1 εi $ p qÁ 4α´2β´1 ď ď ´ ’ν , α β, & i ď %’ β 1.0 β

0.5 0.5

0.5 1.0 1.5 2.0 α 0.5 1.0 1.5 2.0 α

-0.5 -0.5

-1.0 -1.0 (b) (a)

β β

1.0 1.0

0.5 0.5

0.5 1.0 1.5 αα

0.5 1.0 1.5 α -0.5

-0.5 -1.0 n 3 n 5 n 4 “ (d) (c) “ “

Figure 8.1: (a) Region where the 0-form eigenvalue convergence in (8.2.3b) holds

0 i 2β i (max β, 0 α 2); (b) Region where λ X h ν λj X (β α 2, β 0 or β t uă { j p εi q i p 0q ą´ { ě ą α 4, β 0); (c) Region where λ¯p Xi diverges. (α 1 2,α β or 4α 2β 1 0, ´ { ď 1p εi q ą { ě ´ ´ ą α β); (d) Blue region: all eigenvalues diverge. Green region: the form eigenvalues ď diverge and the function eigenvalues converge to 0. Above the red dotted line the volume tends to 0, below it tends to . 8

We now discuss some examples using statements (i’)–(iv’) and equations (8.2.3a”)– (8.2.3c”).

Families of manifolds arising from a sequence of Ramanujan graphs

i i We consider a sequence of discrete Ramanujan graph G i with νi |V G | many p q “ p q i vertices and the associate sequence of equilateral metric graphs X i with all edge p 0q 8.3. Families of manifolds with special spectral properties arising from families of graphs 82 lengths equal to 1 (for a formal deﬁnition see Section 6.3). Then, the (metric) graph Laplacians have a common spectral gap 0,h (see (6.3.7)). Accordingly, we construct p q i a uniform family of graph-like manifolds X i, as described at the beginning of this p εi q section, and assuming conditions (8.3.5) for the parameters εi and ℓi. Consequently, the edge length of the sequence of metric graphs becomes ν´β and the common

2 spectral gap is now dependent on ℓi, i.e., it is given by 0,hi where hi h ℓ (see p q “ { i again (6.3.7)). If we choose α, β from the blue region of picture (d) we have the following. p q

i Proposition 8.3.1. There is a uniform family of graph-like manifolds X i con- p εi q structed as above such that the Hodge Laplacian in all degree forms has an arbitrarily mint2β,2α´1u large spectral gap, i.e., there exists hi h ν such that i Ñ 8

‚ σ ∆Xi 0,hi , p εi q X p q“H

i h ´pn´1qα´β`1 and such that the volume shrinks to 0, more precisely, vol Xεi νi .

In particular, if β 0, i.e, if ℓi 1 for all i, then there exists a common spectral “ “ gap 0,h of the Hodge Laplacian. If, additionally, n 3, then the volume decay can p q “ be made arbitrarily small as α 1 2, i.e., of order ν´2α`1. Œ { i Proof. The proof follows from considerations (i’)–(iv’) above and choosing α, β such p q that α 1 2, β 0 and β α 2 (see Figure 8.1 (d)). We observe that for a sequence ą { ě ď { of Ramanujan graphs, there exists h 0 such that the first non-zero eigenvalue of ą the metric graph Laplacian with unit edge length fulfils λ Xi h for all i, hence 1p 0q ě we can conclude divergence from the first line of (8.2.3b”).

´β We observe that the length scale of the underlying metric graphs is of order νi , ´α but the radius is of order εi ν , which is smaller; hence the injectivity radius of “ i i ´α ´2 2α X is of order εi ν , and the curvature is of order ε ν . εi “ i i “ i It is also possible to construct families of manifolds with ﬁxed volume arising from families of Ramanujan graphs. In order to do so, we need to rescale the metric.

i ´2{n i i We set gi : vol X ,gε gε ) and we consider X : X , gi . Then, the latter “ p p εi i qq i “ p εi q manifold has volume 1. Unfortunately, we cannot have divergence at all degrees at r r r the same time. In fact, for n 3 the conditions are β α 1 2 for divergence of the “ ą ´ { 8.3. Families of manifolds with special spectral properties arising from families of graphs 83 eigenvalues of degree 0, while β α 1 2 is needed for divergence of exact 2-forms. ă ´ { But we can have divergence of 0-forms and higher degree forms separately.

i Proposition 8.3.2. For all n 2 there exists a family of graph-like manifolds X i ě p q of volume 1 with underlying Ramanujan graphs such that the ﬁrst non-zero eigenvalue r on functions (0-forms) diverges.

i ´1{n p1´1{nqα`β{n´1{n Proof. The rescaling factor τi vol X ,gε is of order ν . The “ p p εi i qq i rescaled eigenvalue on functions fulﬁls

λ0 Xi τ ´2λ0 Xi h τ ´2ν2βλ0 Xi h ν2{n´2p1´1{nqpα´βqλ0 Xi , (8.3.6) 1p q“ i 1p εi q i i 1p 0q i 1p 0q and the latterr exponent is positive if and only if β α 1 n 1 . The allowed ą ´ {p ´ q parameters α, β lie inside the triangle 0, 0 , 4, 1 5 n 1 , 2, 1 n 1 such p q p q p ´ q{p p ´ qq p q{p ´ q that λ Xi h ν2{n´δ (see Figure 8.2). The diﬀerence β α approaches its maximum 1p q i ´ on this triangle at the vertex 0, 0 . Hence for any δ 0 there exists α, β inside the r p q ą p q triangle such that λ Xi h ν2{n´δ. 1p q i r n 4 β “ n 3 “ n 2 “

0.5

0.5 1.0 1.5 2.0 α

-0.5

-1.0

Figure 8.2: Parameter region where the rescaled 0-form eigenvalue λ0 Xi diverges, 1p εi q i.e., region where max β, 0 α 2 and λ0 Xi h ν2βλ Xi are both satisﬁed. Above t uă { 1p εi q 1p 0q r the dotted lines, the exponent in (8.3.6) is positive and the eigenvalue diverges. r

In particular, for n 2 Proposition 8.3.2 yields the following corollary. “

i Corollary 8.3.3. There exists a sequence of graph-like surfaces X i of area 1 and p q genus γ Xi with underlying Ramanujan graphs such that the ﬁrst non-zero eigenvalue p q r r 8.3. Families of manifolds with special spectral properties arising from families of graphs 84

i on functions diverges. Moreover, for any δ 0 there exists a sequence X i such ą p q that r λ0 Xi h γ Xi 1´δ. 1p q p q

1 Proof. We have to choose Ye S here,r moreoverr we let the vertex neighbourhood “ be a sphere with k discs removed (as in Example 6.4.1). In this case, the genus of the surface Xi is given by 1 χ Gi where χ Gi is the Euler characteristic of the ´ p q p q graph Gi, and hence r

i i i k k γ X 1 |V G | |E G | 1 νi νi 1 1 νi p q“ ´ p q ` p q “ ´ ` 2 “ ` 2 ´ Ñ 8 ´ ¯ r i as i as k 3 for a Ramanujan graph. In particular, γ X h νi. Ñ 8 ě p q r Family of manifolds arising from a sequence of arbitrary graphs

i i We now consider a sequence of arbitrary discrete graphs G , with νi |V G | p q “ p q Ñ 8 as i and with degrees bounded by k, and the associated sequence of metric Ñ 8 i graphs X i. Then, we construct a sequence of graph-like manifolds with underlying p 0q i metric graph X0 as explained at the beginning of the section assuming (8.3.5) for

εi and ℓi. We show the existence of families of manifolds with constant volume, arbitrarily large form spectrum and convergent function spectrum, hence we do not need that the underlying graphs are Ramanujan. To obtain manifolds with constant

i ´2{n i volume we again set gi vol X gε so that the manifolds X equipped with “ p p εi qq i q the metric gi will have constant volume 1. r r We immediately have the following result. r i Proposition 8.3.4. For all n 3 there exists a family of graph-like manifolds X i ě p q of volume 1, constructed as described above, such that the ﬁrst eigenvalue on exact p- r forms diverges (2 p n 1). Moreover, the ﬁrst non-zero eigenvalue on functions ď ď ´ converges.

Proof. The rescaled eigenvalue on p-forms fulﬁls

λ¯p Xi τ ´2λ¯p Xi τ ´2ν2α´1 h ν2pα´β`1q{n´1, 1p q“ i 1p εi qÁ i i i r 8.3. Families of manifolds with special spectral properties arising from families of graphs 85

(as α β, see (8.2.3c”)) and the latter exponent is positive if and only if β ě ă α n 2 1 . The allowed parameters α, β lie below this line (see Figure 8.3 ´ p { ´ q p q below). For the first non-zero eigenvalue on functions, note first that λ Xi (the first 1p 0q i non-zero eigenvalue of the unilateral metric graph X0) can be bounded from above by π2, this follows immediately from the spectral relation (6.2.3). Therefore, we conclude from (8.3.6) that λ0 Xi 0 as i as β α n 2 1 implies that 1p q Ñ Ñ 8 ă ´ p { ´ q 2 n 2 1 n α β 0. { ´ p ´ qp ´ qă r Actually, comparing the speed of divergence and convergence, we obtain

2 n n2 p 2pn´1q ´ λ¯ Xi ν λ Xi 4pn´1q , 1p qÁ i 1p q conﬁrming again that we cannotr have divergencer for both function and form eigenvalues with our construction.

β n 3 “

n 4 1.0 “

n 5 0.5 “

α 0.5 1.0 1.5

-0.5

-1.0

-1.5

Figure 8.3: Above the dotted lines the p-forms eigenvalues diverge (2 p n 1), ď ď ´ depending on the dimension.

In the special case that our family of graphs consists only of trees, we can modify any given manifold X to become a graph-like manifold (see Remark 6.4.2). In particular, we can show the following corollary. 8.3. Families of manifolds with special spectral properties arising from families of graphs 86

Corollary 8.3.5. On any compact manifold X of dimension n 3, there exists a ě sequence of metrics gi of volume 1 such that the inﬁmum of the (non-zero) function spectrum converges to 0, while the exact p-form eigenvalues (2 p n 1) diverge. ď ď ´ Appendix A

Geometry of T kM

We here describe the space T kM introduced in Chapter 2. We have already noticed that vectors on T kM can be decomposed into horizontal and vertical components, resembling the structure of the tangent spaces of TM. In this Appendix we will analyse the Lie brackets and the covariant derivative of horizontal and vertical vectors, and we will look at curvature properties of T kM in relation to the curvature of the base manifold M. The reader will ﬁnd it useful to compare these results with the ones in [Dom62, GuKa02,KS05] and references therein. In [Dom62,GuKa02], the authors describe the geometry of TM equipped with the Sasaki metric, while in [KS05], the authors describe the geometry of the linear frame bundle LM over a manifold equipped with a Sasaki-type metric. In both articles, the authors use local coordinates in their geometric descriptions of TM and LM. Moreover, they also discuss other types of metric on TM and LM.

Let M,g be a compact n-dimensional manifold with tangent bundle TM, and p q let π : TM M be the canonical projection. We remind the reader that for ÝÑ k 1,...,n, T kM is deﬁned as “

k T M f v1,...,vk TpM ... TpM π vi p i 1,...,k , “ pPMt “ p q P ˆ ˆ | p q“ @ “ u ď and that there is a canonical projection πk : T kM M such that πk f p if ÝÑ p q “ vi TpM for all i 1,...,k. P “ 87 A.1. Horizontal and vertical lifts 88

T kM is a manifold of dimension n nk and its tangent space at a point f is given ` by k Tf T M TpM ... TpM . “ ˆ ˆ pk`1q´times In fact, any tangent vector on T kM isloooooooooomoooooooooon described as

1 d k D D X 0 π X t ; V1 t ,..., Vk t , p q“ dt t“0p ˝ qp q dt t“0 p q dt t“0 p q ´ ˇ ˇ ˇ ¯ ˇ k ˇ ˇ k where X V ,...,Vk :ˇ ε,ε T Mˇ is a curve on ˇT M. In view of this, “ p 1 q p´ q ÝÑ and having in mind the decomposition of vectors on TM into horizontal and vertical h component, we consider any vector u on T kM as the sum of u u ;0,..., 0 and “ p 0 q v u 0; u ,...,uk . Moreover, we deﬁne “ p 1 q

h k Hf u u Tf T M T k M and “ t | P u– π pfq f T kM Pď v k Vf u u Tf T M T k M ... T k M “ t | P u– π pfq ˆ ˆ π pfq fPT kM ď k´times to be the horizontal and vertical distributionsloooooooooooooomoooooooooooooon at the point f. Consequently,

k Tf T M Hf Vf . “ `

We also remind the reader that we equip T kM with the Sasaki-type metric

gf u,w gp u0,w0 gp ui,wi , p q“ p q` i“1 p q ÿ k for every u u ; u ,...,uk and w w ; w ,...,wk vectors in Tf T M with “ p 0 1 q “ p 0 1 q πk f p. Hence, horizontal and vertical components are pairwise orthogonal. p q“

A.1 Horizontal and vertical lifts

k k k Let f T M with π f p, and let u u ; u ,...,uk Tf T M. For all P p q “ “ p 0 1 q P i 0,...,k, we deﬁne the map “

k πi : Tf T M TpM such that πi u ui. ÝÑ p q“ A.1. Horizontal and vertical lifts 89

k Definition A.1.1. Let w TpM. The horizontal lift of w to a point f T M is P P h k h h the unique vector w Tf T M such that π w w and πi w 0 for i 0, i.e., P 0p q“ p q“ ‰ wh w;0,..., 0 . “ p q The horizontal lift of a vector field X on M is the unique vector field Xh on T kM

h h k such that π X X p TpM and πi X 0 for i 0 for all f T M with 0p q “ p q P p q “ ‰ P πk f p. p q“

k Definition A.1.2. Let w TpM. The j-th vertical lift of w to a point f T M P P v v v v is the unique vector w such that πj w w and πi w 0 for i j, i.e., w j p j q “ p j q “ ‰ j “ 0;0,..., 0, w , 0,..., 0 . p q j-th place j X M Xv The -thloomoon vertical lift of a vector field on is the unique vector field j on k v v k T M such that πj X X p TpM and πi X 0 for i j for all f T M p j q “ p q P p j q “ ‰ P with πk f p. p q“ We observe that the maps w wh and w wv for all j are vector isomorphisms ÞÑ ÞÑ j between TpM and Hf and between TpM and the j-th copy of TpM in Vf , respectively. k Therefore, the horizontal and vertical component of any vector z Tf T M can be P interpreted as horizontal and vertical lift, i.e., we have

k h v h v z z z z0 zi z0; z1,...,zk . “ ` “ ` i“1 “ p q ÿ Therefore, it is suﬃcient to look at the horizontal and vertical lifts to recover the behaviour of the horizontal and vertical component of a vector on T kM.

8 k We also note that for every h C M and every w Tf T M, we have P p q P

wh h πk w h and wv h πk 0, (A.1.1) p ˝ q“ 0p q j p ˝ q“ PC8pT kMq loomoon while for every H C8 T kM , we have P p q

h d w H f H fw t , (A.1.2) p qp q“ dt t“0 p p qq d ˇ v ˇ v wj H f H f tJ wj , (A.1.3) p qp q“ dt t“0ˇ p ` p qq ˇ k k ˇ where J : T T M T M is such thatˇ J u J u ; u ,...,uk u ,...,uk . ÝÑ p q “ pp 0 1 qq “ p 1 q h v In fact, we can think of w and wj to be the generators of the local 1-parameter A.2. Lie Brackets 90

v groups ϕt f fw t and ϕt f f tJ w v ,...,vj ,vj tw,vj ,...,vk p q “ p q p q “ ` p j q “ p 1 ´1 ` `1 q for f v1,...,vk . “ p q r We also point out that (A.1.2) and (A.1.3) correspond to the deﬁnition of horizontal and j-th vertical gradient of a smooth function on T kM as introduced in Section 2.3.

A.2 Lie Brackets

h v h v Proposition A.2.1. Let X,Y be vector ﬁelds on M and consider X ,Xj ,Y ,Yj their horizontal and j-th vertical lifts for j 1,...,k. Then, for all f v ,...,vk “ “ p 1 q P T kM and πk f p, we have p q“ k h h h v X ,Y f X,Y p Rp X,Y vi r sp q“pr sp qq ´ p p q qi i“1 (A.2.4) ÿ X,Y f ; Rp X,Y v ,..., Rp X,Y vk , “ r sp q ´ p q 1 ´ p q ´ ¯ h v v X ,Y f ∇X p Y 0;0,..., ∇X p Y , 0 ..., 0 , (A.2.5) r j sp q “ p p q qj “ p p q q j-th place Xv,Y v f 0 i, j loomoon1 ...,k, (A.2.6) r j i sp q“ @ “ where Rp is the Riemannian curvature tensor on M evaluated at the point p.

k Proof. Let J : Tf T M TpM ... TpM be such that J u ; u ,...,uk ÝÑ ˆ ˆ p 0 1 q “ h v u ,...,uk . Let ϕs, ϕs and ϕt, ϕt be local 1-parameter groups associated to X , X p 1 q j h v and Y , Yj , respectively, i.e., r r r r

k k k k r r ϕs : R T M T M ϕs : R T M T M ˆ ÝÑ ˆ ÝÑ v s,f fXppq s s,f f sJ Xj f , p q ÞÑ p q pr q ÞÑ ` p p qq h v the same for ϕt, ϕt associated to Y , Yj , respectively. r ´1 Note that ϕ‚ ϕ´‚. r“ r r Using [KN63, Proposition 1.9], we have

h h 1 h h X ,Y f lim Y f dϕs Y f . r sp q“ sÑ0 s p q ´ p p qqp q ` ˘ A.2. Lie Brackets 91

Then,

h d ´1 dϕs Y f ϕs ϕt ϕs f p p qqp q“ dt t“0p ˝ ˝ qp q d ˇ r ˇ ϕs ϕ´s f Y ppq t “ dtˇt“0 pp p qq qp q d ˇ ˇ fY ppq t Xpqq s , “ dtˇt“0p p qq p q ˇ ˇ where q c t , and c is the geodesic withˇ starting vector Y p . “ p q p q We now deﬁne the variation H s,t fY p t s H s,t ,...,Hk s,t p q “ p qp q Xpqqp q “ p 1p q p qq where Hj s,t vj Y ppq t Xpqq s . Then,` ˘ p q “ pp q p qq p q

h k D D dϕs Y f B H s,t B π H s,t ; H1 s,t ,..., Hk s,t . p p qqp q“ t t“0 p q“ t t“0p ˝ qp q dt t“0 p q dt t“0 p q B ˇ ´B ˇ ˇ ˇ ¯ ˇ ˇ ˇ ˇ Since Y h f ˇY p ;0,..., 0 ˇ B πk H 0,tˇ ;0,..., 0 , we haveˇ p q “ p p q q“ Bt t“0p ˝ qp q 1 ` ˇ ˘ Xh,Y h f lim B πk Hˇ 0,t B πk H s,t ; r sp q“ sÑ0 s t t“0p ˝ qp q´ t t“0p ˝ qp q ´ ´B ˇ 1 D B ˇ 1 D¯ ˇ lim H1 s,tˇ ,..., lim Hk s,t ˇ ´ sÑ0 s dt t“0 p ˇ q ´ sÑ0 s dt t“0 p q D D ˇ D D ˇ ¯ X,Y p ; ˇ H1 s,t ,..., ˇ Hk s,t . “ r sp q ´ds s“0 dt t“ˇ0 p q ´ds s“0 dt tˇ“0 p q ´ ˇ ˇ ˇ ˇ ¯ ˇ ˇ ˇ ˇ Now, ˇ ˇ ˇ ˇ

D D k k Hi s,t R B π H 0,t , B π H s, 0 Hi 0, 0 ´ds s“0 dt t“0 p q“ t t“0p ˝ qp q s s“0p ˝ qp q p q ˇ ˇ ´B ˇ B ˇ D D ¯ ˇ ˇ ˇ ˇ Hi s,t ˇ ˇ ˇ ˇ ´dt t“0 ds s“0 p q ˇ ˇ D ´ ˇ pviˇqY ppqptq“0 dtˇ t“0 ˇ loooooooooooomoooooooooooon Rp Y,X vi. ˇ “ p q ˇ

Hence,

h h X ,Y f X,Y f ; Rp X,Y v ,..., Rp X,Y vk , r sp q“pr sp q ´ p q 1 ´ p q q which proves (A.2.4).

To prove (A.2.5), we proceed as before using the local 1-parameter groups ϕs and h v ϕt generating X and Yj , respectively. Then, we have

r rh v 1 v v r X ,Yj f lim Yj f dϕs Yj f r sp q“ sÑ0 s p q ´ p qp qp q ` ˘ A.3. Covariant Derivative 92 and

v d ´1 dϕs Yj f ϕs ϕt ϕs f p qp qp q“ dt t“0p ˝ ˝ qp q d ˇ ˇ v ϕs ϕ´rs f tJ Yj ϕ´s f “ dtˇt“0 p p q` p p p qqqq vˇ Y ˇ ϕ s f k s . “ j ˇp ´ p qqXpπ pϕ´spfqqqp q Therefore,

h v 1 v v X ,Yj f lim Yj ϕ0 f Yj ϕ´s f Xpπkpϕ pfqqq s r sp q“ sÑ0 s p p qq ´ p p qq ´s p q ` ˘ Z f ; Z f ,...,Zk f “ p 0p q 1p q p qq where Zi f 0 for all i j and p q“ ‰

1 k k Zj f lim Y π ϕ0 f Y π ϕ´s f Xpπkpϕ pfqqq s ∇XppqY. p q“ sÑ0 sp p p p qqq ´ p p p qq ´s p qq “

Finally, we consider the local 1-parameter groups ϕs, ϕt to prove (A.2.6). It is easy to see that ϕs and ϕt commute. Therefore, by [KN63, Corollary 1.11] we conclude r r v v Xi ,Yj 0 for all i, j 1,...,k. r s“ r r “

A.3 Covariant Derivative

Let ∇, ∇ be the Levi-Civita connection on M,g and T kM, g respectively. We re- p q p q call that for any V,U,W vector ﬁelds on T kM, Kozul formula holds [Sak96, Equation 1.13, p. 28].

1 g ∇V U, W V g U, W U g V,W W g V, U f p q“ 2 p f p qq ` p f p qq ´ p f p qq ` g V, U, W g U, V,W g W, U, V (A.3.7) ´ f p r sq ´ f p r sq ´ f p r sq ˘ Using the formula above, we are able to compute the covariant derivatives of horizontal and vertical lifts.

Proposition A.3.1. Let X,Y be two vector ﬁelds on M, Xh,Y h be their horizontal lifts on T kM and Xv,Y v be their j-th and i-th vertical lifts for i, j 1,...,k. Then, j i “ k k for all f v ,...,vk T M with π f p, we have “ p 1 q P p q“

v ∇Xv f Y 0, (A.3.8) j p q i “ A.3. Covariant Derivative 93

k h h 1 v ∇ h Y ∇X p Y Rp X,Y vi X pfq “p p q q ´ 2 p q i i“1 (A.3.9) 1ÿ ` ˘ 1 ∇X p Y ; Rp X,Y v ,..., Rp X,Y vk , “ p q ´2 p q 1 ´2 p q ´ ¯ v 1 h v ∇XhpfqYi Rp vi,Y X ∇XppqY i “ 2 p q ` p q (A.3.10) `1 ˘ Rp vi,Y X;0,..., 0, ∇X p Y, 0,..., 0 , “ p2 p q p q q

h 1 h 1 ∇Y v f X Rp vi,Y X Rp vi,Y X;0,..., 0 . (A.3.11) i p q “ 2p p q q “ 2 p q ´ ¯ h v k Proof. Let W W W W ; W ,...,Wk be a vector ﬁeld on T M. In order to “ ` “ p 0 1 q understand the horizontal and vertical components of the vectors in (i)–(iv), we take h v v their inner products against W and the l-th component of W , denoted by W l. In what follows, we will make use of the deﬁnition of g, Proposition A.2.1 and equations (A.1.1) and (A.3.7). We have

v v v v v 1 v v v v v v g ∇Xv Y , W l X g Y , W l Y g X , W l W l g X ,Y f p j i q“ 2 j p f p i qq ` i p f p j qq ´ p f p j i qq v v v ` v v v v v v grf Xi , Yj , W l g Yj , Xj , W l g W l, Yi ,Xj 0, ´ p r sq ´ p r sq ´ p r sq “ ˘ and

h h h h v 1 v v v v v v g ∇Xv Y , W X g Y , W Y g X , W W g X ,Y f p j i q“ 2 j p f p i qq ` i p f p j qq ´ p f p j i qq ` h h h g Xv, Y v, W g Y v, Xv, W g W, Y v,Xv ´ f p i r j sq ´ p j r j sq ´ p r i j sq 1 ˘ W gp Xj,Yi gp Xj, ∇W Yi gp Yi, ∇W Xj . “ 2 ´ 0p p qq ` p q` p q ` ˘ If i j, then each term of the above formula is zero due to the deﬁnition of g. If ‰ i j, then the above sum is zero due to the Riemannian property of ∇. This proves “ (A.3.8). Now, we prove (A.3.9). As before,

h h h h 1 h h h h h h g ∇ h Y , W X g Y , W Y g X , W oshW g X ,Y f p X q“ 2 p f p qq ` p f p qq ´ p f p qq ` h h h g Xh, Y h, W g Y h, Xh, W g W, Y h,Xh ´ f p r sq ´ f p r sq ´ f p r sq ˘ A.3. Covariant Derivative 94

1 X gp Y,W Y gp X,W W gp X,Y “2 p p 0qq ` p p 0qq ´ 0p p qq ` g X, Y,W gp Y, X,W gp W , Y,X ´ pp r 0sq ´ p r 0sq ´ p 0 r sq

gp ∇X Y,W , ˘ “ p 0q and

v v v v h 1 h h h h h h g ∇ h Y , W l X g Y , W l Y g X , W l W l g X ,Y f p X q“ 2 p f p qq ` p f p qq ´ p f p qq v v v ` h h h h h h g X , Y , W l g Y , X , W l g W l, Y ,X ´ f p r sq ´ f p r sq ´ f p r sq v 1 h h ˘ g W l, Y ,X “´2 f p r sq 1 gp Wl,R Y,X vl . “ 2 p p q q

h 1 Hence, the l-th vertical component of ∇ h Y is R Y p ,X p vl for all l X pfq 2 p p q p qq “ 1,...k, and so (A.3.9) is proved. Now, we look at (A.3.10). Again,

h h h h v 1 h v v h h v g ∇ h Y , W X g Y , W Y g X , W W g X ,Y f p X i q“ 2 p f p i qq ` i p f p qq ´ p f p i qq ` h h h g Xh, Y v, W g Y v, Xh, W g W, Y v,Xh ´ f p r i sq ´ f p i r sq ´ f p r i sq h 1 v h ˘ gf Y , X , W “´2 p i r sq 1 gp Yi,R X,W vi “ 2 p p 0q q 1 gp R vi,Yi X,W , “ 2 p p q 0q and

v v v v v 1 h v v h h v g ∇ h Y , W l X g Y , W l Y g X , W l W l g X ,Y f p X i q“ 2 p f p i qq ` i p f p qq ´ p f p i qq v v ` h v v h v h g X , Y , W l g Y , X , vWl g W l, Y ,X ´ f p r i sq ´ f p i r sq ´ f p r i sq v v 1 h v v h v h ˘ X g Y , W l g Y , X , vWl g W l, Y ,X “ 2 p f p i qq ´ f p i r sq ´ f p r i sq 1` ˘ δil X gp Y,Wl gp Y, ∇X Wl gp Wl, ∇X Y “ 2 p p q´ p q` p q ` ˘ δilgp ∇X Yi,Wl , “ p q where the last equality is due to the Riemannian property of ∇. Therefore, (A.3.10) is proved since the only non-zero component is the i-th (l i). “ A.3. Covariant Derivative 95

Finally, we look at (A.3.11). Using (A.2.5) and (A.3.10), we obtain.

h v h v ∇Y v f X ∇ h Y X ,Y f i p q “ X pfq i ´r i sp q h 1 v v Rp vi,Y X ∇X p Y ∇X p Y “ 2 p q ` p p q qi ´ p p q qi ´1 ¯h Rp vi,Y X , “ 2 p q ´ ¯ which concludes the proof.

We analyse the Levi-Civita connection of the horizontal and i-th vertical lift of a semi-basic vector ﬁeld. We remind the reader that a semi-basic vector ﬁeld is a map

k k F : T M TM such that F f TpM if π f p (see also Section 2.3). ÝÑ p q P p q“ Deﬁnition A.3.2. Let F : T kM TM be a semi-basic vector ﬁeld. The horizontal ÝÑ and i-th vertical lift of F are the maps

h F h : T kM T T kM, F h f F f ÝÑ p q“ p q ` ˘ and v F v : T kM T T kM, F v f F f i ÝÑ i p q“ p q i ` ˘ k We now consider a very special semi-basic vector field. We define Pi : T M ÝÑ TM such that Pi f vi for every f v ,...,vk , i.e., Pi is the projection of f on p q“ “ p 1 q the i-th component of T kM, and we consider G : TM TM, an endomorphism on ÝÑ k TM. Then, H G Pi is a semi-basic vector field on T M. We have the following “ p ˝ q result.

h v Proposition A.3.3. Let H be the semi-basic vector ﬁeld deﬁned above. Let X ,Xj be the horizontal and j-th vertical lift of X X M and let ∇ be the Levi-Civita P p q k connection on T M. Then, for all f v ,...,vk , we have “ p 1 q h h ∇ h H ∇ h H V , (A.3.12) X pfq “ X pfqp ˝ q v v ∇ h H ∇ h H V , (A.3.13) X pfq i “ X pfqp ˝ qi v v v ∇Xv f H H J X f , (A.3.14) j p q i “ p p j p qqq i h v ` 1 ˘ h ∇Xv f H H J X f Rp vj,X H f , (A.3.15) j p q “ p p j p qqq ` 2 p q p q ` ˘ where V V ,...,Vk is a realization of f, i.e., f V f V p ,...,Vk p , and “ p 1 q “ p q “ p 1p q p qq k J : Tf T M TpM ... TpM is such that J u ; u ,...,uk u ,...,uk . ÝÑ ˆ ˆ p 0 1 q “ p 1 q A.3. Covariant Derivative 96

h Proof. We consider the curve ϕt generating X , i.e., ϕt f fX p t . Then, p q“ p qp q

h d h ∇XhpfqH H fXppq s “ ds s“0 p p qq d ˇ h ˇ H V fXppq s “ dsˇs“0 p p p qqq d ˇ h ˇ H V ϕs f “ dsˇs“0pp ˝ q˝ p qqq ˇ h ∇ ˇh H V , “ Xˇ pfqp ˝ q which proves (A.3.12). Equation (A.3.13) follows from the same considerations.

n B To prove (A.3.14) and (A.3.15), we ﬁrst observe that for vi dxα vi , “ α“1 p q Bxα p we have ˇ ř ˇ n n

H f G Pi f G vi aα p G B dxα vi G B . p q “ p ˝ qp q“ p q“ p q xα p “ p q xα p α“1 ´B ˇ ¯ α“1 ´B ˇ ¯ ÿ ˇ ÿ ˇ Therefore, ˇ ˇ n v v k Hi f dxα vi G B π f x i p q“ α“1 p q p ˝ α ˝ p q ÿ ´ B ¯ and n h h k H f dxα vi G B π f . x p q“ α“1 p q p ˝ α ˝ qp q ÿ ´ B ¯ Hence,

n v k v ∇Xv f H ∇Xv f dxα G B π j p q i j p q x i “ α“1 p ˝ α ˝ q ÿ B n ` ˘ v v k k v Xj dxα f G B π f dxα vi ∇Xvpfq G B π i x i j x “ α“1 p q p q p ˝ α ˝ qp q ` p q p ˝ α ˝ q ÿ ´ B ¯ B ` ˘ “0 by (A.2.6)

v v looooooooooooomooooooooooooon Since X f is generated by ϕt f f tJ X f , we have j p q p q“ ` p j p qq

v v X dxα f X f dxα Pi f j p qp q“ j p q p p p qqq `d ˘ v dxα Pi f tJ Xj f “ dt t“0 p ` p p qqq (A.3.16) d ˇ ˇ ` ˘ v dxα Pi f tdxα Pi J Xj f “ dtˇt“0 p p qq ` p p p p qqqq ˇ v dxˇα Pi J X f , “ ˇ p p j p qqq ` ˘ where the third and fourth equality are due to the fact that dxα and Pi are linear. A.4.TheRiemanniancurvaturetensor 97

Hence, we conclude

n v v v k v v ∇XvpfqHi dxα Pi J Xj f G B π f H J Xj f , j x i i “ α“1 p p p qqq p ˝ α ˝ qp q “ p p p qqq ÿ ` ˘´ B ¯ ` ˘ which proves (A.3.8). We proceed in the same way to prove (A.3.11).

n h k h ∇Xv f H ∇Xv f dxα G B π j p q j p q x “ α“1 p ˝ α ˝ q ÿ B n ` ˘ h v k k h X dx f G π f dx v v G π j α B α i ∇Xj pfq B “ p q p q p ˝ xα ˝ qp q ` p q p ˝ xα ˝ q α“1 B B ÿ ` ˘ ´ h¯ v h 1 H J X f Rp vj,X H f , “ p p j p qqq ` 2 p q p q ´ ¯ where the last equality is due to (A.3.16) and (A.3.11).

A.4 The Riemannian curvature tensor

Let R and R be the Riemannian curvature tensor on T kM and M, respectively, and let Rf ,Rp be their evaluations at the point f and p, respectively. For any V, W, U X T kM , we have P p q

Rf V,W U ∇V f ∇W f U ∇W f ∇V f U ∇ V,W f U. (A.4.17) p q “ p q p q ´ p q p q ´ r sp q Proposition A.4.1. Let X,Y X M and consider their horizontal and vertical P p q k k lifts, denoted as usual. Let f v ,...,vk T M with π f p. Then, “ p 1 q P p q“

v v v Rf X ,Y Z 0 i,j,l 1,...,k, (A.4.18) p i j q l “ @ “ h h v v 1 1 Rf X ,Y Z Rp vj,Z Rp vl,Z X δjlRp Y,Z X , (A.4.19) p j q l “ ´ 4 p qp p q q´ 2 p q ´ ¯

v v h 1 Rf X ,Y Z δijRp X,Y Z Rp vi,X Rp vj,Y Z p i j q “ p q ` 4 p qp p q q ´ 1 h Rp vj,Y Rp vi,X Z , (A.4.20) ´ 4 p qp p q q ¯

h v h v h 1 1 Rf X ,Yj Z ∇XppqR vj,Y Z Rp X,Z Y p q “ 2 p q ` 2 p q j ´ ¯ ´ ¯ 1 k v Rp X,Rp vj,Y Z vi , (A.4.21) 4 i ´ i“1 p p q q ÿ ´ ¯ A.4.TheRiemanniancurvaturetensor 98

h v h h v 1 Rf X ,Y Zl ∇XppqR vl,Z Y ∇Y ppqR vl,Z X Rp X,Y Z p q “ 2 p q q´ p q ` p q l ´ k ¯ ´ ¯ 1 v Rp Y,R vl,Z X vi Rp X,R vl,Z Y vi , (A.4.22) 4 i i“1 p p q q ´ p p q q ÿ ´ ¯

1 k R Xh,Y h Zh R X,Y Z R v ,R X,Z v Y p 4 p i i p q “ p q ` i“1 p p q q ´ ÿ h Rp vi,R Y,Z vi X 2Rp vi,R X,Y vi Z ´ p p q q ` p p q q 1 k ¯ v ∇ZppqR X,Y vi . (A.4.23) 2 i ´ i“1 p q ÿ ´ ¯ Proof. We will make use of (A.4.17) to prove the proposition. Equation (A.4.18) is an easy consequence of (A.4.17), (A.2.6), and (A.3.8).

Before proving the remaining equations, we observe that the map F f F v ,...,vk p q“ p 1 q“ R vi,X p Y p is a semi-basic vector ﬁeld for any X,Y X M and any index i. In p p qq p q P p q particular, Lemma A.3.3 applies to such a map. We now prove (A.4.19). Using (A.2.5), Proposition A.3.1 and Lemma A.3.3, we have

h v v v v v Rf X ,Y Z ∇Xh f ∇Y vpfqZ ∇Y vpfq∇Xh f Z ∇ Xh,Y v f Z p j q l “ p q j l ´ j p q l ´ r j sp q l v ∇Y v f ∇ h Z “´ j p q X pfq l 1 h v ∇Y v f R vl,Z X ∇Y v f ∇X p Z “´2 j p qp p q q ´ j p qp p q ql “0 1 1 h δjlRp Y,Z X Rp looooooooomooooooooonvj,Y Rp vl,Z X . “´2 p q ´ 2 p qp p q ´ ¯ To prove (A.4.20), we use the Bianchi Identity and (A.4.19). Then,

v v h h v v v h v Rf X ,Y Z f Rf Z ,X Y Rf Y ,Z X p i j q qp q“´ p i q j ´ p j q i h v v h v v Rf Z ,X Y Rf Z ,Y X “´ p i q j ` p j q i 1 δijRp X,Y Z Rp vi,X Rp vj,Y Z “ p q ` 4 p qp p q q 1´ h Rp vj,Y Rp vi,X Z . ´ 4 p qp p q q ¯ Now,

h v h h h h Rf X ,Y Z ∇Xh f ∇Y vpfqZ ∇Y vpfq∇Xh f Z ∇ Xh,Y v f Z p j q “ p q j ´ j p q ´ r j sp q A.4.TheRiemanniancurvaturetensor 99

k 1 h h 1 v ∇ h R vj,Y X ∇Y v f ∇X p Y ∇Y v f R X,Z vi 2 X pfq j p q p q 2 j p q i “ p p q q ´ p q ` i“1 p p q q ÿ h ∇ ∇ V v Z ´ p Xppq qj k v 1 h 1 1 h ∇XppqR vj,Y Z , Rp X,Rp vj,Y Z vi Rp vj,Y ∇XppqZ 2 4 i 2 “ p p q q ´ i“1 p p q q ´ p p qp qq ÿ ´ ¯ 1 h 1 v Rp vj, ∇X p Y Z Rp X,Z Y ` 2p p p q q q ` 2p p q qi 1 h 1 v 1 k v ∇XppqR vj,Y Z Rp X,Z Y Rp X,Rp vj,Y Z vi , 2 2 j 4 i “ p q ` p q ´ i“1 p p q q ´ ¯ ´ ¯ ÿ ´ ¯ which proves (A.4.21). We again use the Bianchi Identity to prove (A.4.22).

h h v v h h h v h Rf X ,Y Z Rf Z ,X Y Rf Y ,Z X p q l “´ p l q ´ p l q h v h h v h Rf X ,X Y Rf Y ,Z X “ p l q ´ p l q h 1 v ∇X p R vl,Z Y ∇Y p R vl,Z X Rp X,Y Z “ 2 p q p q ´ p q p q ` p p q ql ´ k ¯ 1 v Rp Y,Rp vl,Z Y vi Rp X,Rp vl,Z Y vi . 4 i ` i“1 p p q q ´ p p q q ÿ ´ ¯ Finally, we prove (A.4.23).

h h h h h h Rf X ,Y Z ∇ h ∇ h Z ∇ h ∇ h Z ∇ h h Z p q “ X pfq Y pfq ´ Y pfq X pfq ´ rX ,Y spfq k k h 1 v h 1 v ∇ h ∇Y p Z Rp Y,Z vi ∇ h ∇X p Z Rp X,Z vi “ X pfq p q q ´ 2 p p q qi ´ Y pfq p q q ´ 2 p p q qi ´ i“1 ¯ ´ i“1 ¯ ÿ k ÿ h h h Z v Z ∇prX,Y sppqq ∇pRppX,Y qviqi ´ ` i“1 ÿ 1r k ∇ ∇ Z h R X, ∇ Z v v ∇ ∇ Z h Xppq Y ppq 2 p Y ppq i i Y ppq Xppq “ p q ´ i“1p p q q ´ p q ÿ 1 k 1 k R Y, ∇ Z v v R v ,R Y,Z v X h 2 ∇ R Y,Z v v 2 p Xppq i i 4 p i p i Xppq i i ` i“1p p q q ´ i“1 p p p q q q ` p p q q ÿ ÿ ´ ¯ 1 k R v ,R X,Z v Y h 2 ∇ R X,Z v v ∇ Z h 4 p i p i Y ppq i i rX,Y sppq ` i“1 p p p q q q ` p p q q ´ p q ÿ ´ ¯ 1 k 1 k R X,Y ,Z v v R v ,R X,Y v Z h 2 p i i 2 p i p i ` i“1p pr s q q ` i“1p p p q q q ÿ ÿ A.5. Curvature 100

1 k R X,Y Z R v ,R X,Z v Y R v ,R Y,Z v X p 4 p i p i p i p i “ p q ` i“1 p p q q ´ p p q q ´ ÿ “ h 1 k v 2Rp vi,Rp X,Y vi Z ∇ZppqR X,Y vi . 2 i ` p p q q ´ i“1 p q ‰¯ ÿ ´ ¯

A.5 Curvature

Let K, Ric, S be the sectional curvature, the Ricci curvature and the scalar curvature

k k on T M, respectively. For any v,w Tf T M we have P g R v,w w,v K v,w f p p q q , (A.5.24) pt uq “ v 2 w 2 g v,w } } } } ´ f p q npk`1q

Ric v,v K v,ei , (A.5.25) p q“ i“1 pt uq ÿ npk`1q

S Ric ei,ei , (A.5.26) “ i“1 p q ÿ where v,w is the plane spanned by v and w, and where e ,...,en k is an or- t u 1 p `1q k thonormal basis of Tf T M.

Proposition A.5.1. Let X,Y X M be two unitary vector ﬁelds and consider their P p q horizontal and vertical lifts as usually denoted. The sectional curvature K of T kM with respect to the Sasaki-type metric satisﬁes the followings.

K Xv f ,Y v f 0 i, j 1,...,k, (A.5.27) pt i p q j p quq “ @ “

h v 1 2 K X f ,Y f R vi,Y p X p , (A.5.28) pt p q j p quq “ 4} p p qq p q} 3 k K Xh f ,Y h f K X p ,Y p R X,Y v 2. (A.5.29) 4 p i pt p q p quq “ pt p q p quq ´ i“1 } p q } ÿ h h v v Proof. To prove the proposition we plug in (A.5.24) X ,Y ,Xi ,Yj . Equation (A.5.27) holds due to (A.4.18). Since, X,Y are unitary, then Xh 2 Y v 2 Y h 2 1. Using (A.4.19), we } } “} j } “} } “ have A.5. Curvature 101

K Xh f ,Y v f g R Xh,Y v Y v,Xh pt p q j p quq “ f p p j q j q 1 1 gp R vj,Y R vj,Y X ,X gp R Y,Y X,X “´4 p p qp p q q q´ 2 p p q q 1 2 Rp vi,Y X , “ 4} p q } which proves (A.5.28). Finally, using (A.4.22), we obtain

K Xh f ,Y h f g R Xh,Y h Y h,Xh pt p q p quq “ f p p q q

gp R X,Y Y,X “ p p q q 1 k g R v ,R X,Y v Y,X g v ,R Y,Y v X,X 4 p i i p i i ` i“1 p p p q q q´ p p q q q ÿ ´ 2g Rp vi,R X,Y vi Y,X ` p p p q q q 3 k ¯ K X p ,Y p R X,Y v 2, 4 p i “ pt p q p qu ´ i“1 } p q } ÿ which proves (A.5.29).

From the above proposition, we derive some relations between the curvature of T kM and M.

Proposition A.5.2. Let M,g be a Riemannian manifold and let T kM, g be the p q p q bundle of k-frames on M equipped with a Sasaki-type metric. Then, T kM is ﬂat if and only if M is.

Proof. Statement i is a direct consequence of Proposition A.5.1 or of Proposition p q (A.4.1).

Proposition A.5.3. Let M,g and T kM, g be as in the above proposition. Then, p q p q the following statements are true.

(i) If T kM has bounded sectional curvature, then it is ﬂat,

(ii) If T kM has bounded sectional curvature, then M is ﬂat. A.5. Curvature 102

Proof. By contradiction, we assume that T kM is not flat. Then, M is not flat by Proposition A.5.2. Hence, there exist a point p M and a pair of orthonormal P vectors u ,u TpM such that Rp u ,u V 0 for some V vector field on M. 1 2 P p 1 2q ‰ h h 3 k 2 h h Then, K u ,u K u ,u Rp u ,u vi for u ,u the horizontal pt 1 2 uq “ pt 1 2uq ´ 4 i“1 } p 1 2q } 1 2 k lifts of u1,u2 at the point f v1,...,vřk such that π f p. Since the set of vi “ p q p q “ satisfying this condition is unbounded, so is the set of f which has vi has one of the components. In a similar way, we show that K uh,uh is unbounded from above pt 1 2 uq using the (A.5.28). This proves statement i . p q Statement ii is a consequence of statement i and of Proposition A.5.2. p q p q We now look at the Ricci and scalar curvature of T kM. We fist observe that given

h h e1,...,en an orthonormal basis for TpM, then e1 ,...,en is an orthonormal basis for v v Hf and e ,... en for i 1,...,k is an orthonormal basis for Vf . p 1qi p qi “

h h v Corollary A.5.4. Let e ,...,e be an orthonormal basis for Hf , ej for i 1,...,k 1 k p qi “ h v and j 1,...,n be an orthonormal basis for Vf as described above. Let u ,u be “ l the horizontal and l-th vertical lifts of u TpM at the point f v ,...,vk with P “ p 1 q πk f p. Then, p q“ 1 k n Ric uh,uh Ric u,u R e ,v u 2, (A.5.30) 2 p j i p q“ p q´ i“1 j“1 } p q } ÿ ÿ 1 n Ric uv,uv R v ,u e 2. (A.5.31) l l 4 l i p q“ i“1 } p q } ÿ Proof. We apply Proposition A.5.1. Then,

n n k h h h h h v Ric u ,u K u ,ej K u , ej i p q“ j“1 pt uq ` j“1 i“1 pt p q uq ÿ ÿ ÿ k 3 k 1 k n K u,e R u,e v 2 R v ,e u 2 j 4 p j l 4 p i j “ j“1 pt uq ´ l“1 } p q } ` i“1 j“1 } p q } ÿ ` ÿ ˘ ÿ ÿ n k 1 3 Ric u,u R v ,e u 2 R u,e v 2. 4 p i j 4 p j i “ p q` j“1 i“1 } p q } ´ } p q } ÿ ÿ n α n We now observe that for vi v eα and u ulel, we have “ α“1 i “ l“1 ř ř A.5. Curvature 103

n 2 α 2 Rp u,ej vi ulvi Rp el,ej eα j`1 } p q } “ j } l,α p q } ÿ ÿ ÿ α β uluqvi vi gp R el,ej eα,es gp R eq,ej eβ,es “ j,q,l,α,β,s p p q q p p q q ÿ α β uluqvi vi gp R es,eα el,ej gp R eβ,es eq,ej “ j,q,l,α,β,s p p q q p p q q ÿ α β uluqvi vi gp R ej,eα el,R ej,eβ eq “ j,q,l,α,β p p q p q q ÿ 2 Rp ej,vi u . “ j } p q } ÿ Therefore, 1 k n Ric uh,uh Ric u,u R e ,v u 2, 2 p j i p q“ p q´ i“1 j“1 } p q } ÿ ÿ which proves (A.5.30). Equation (A.5.31) is an easy consequence of (A.5.28).

h h v Corollary A.5.5. Let e ,...,e and let ej for i 1,...,k and j 1,...,n be as 1 k p qi “ “ above. Let f v ,...,vk , then “ p 1 q 1 n k S S R e ,v e 2 (A.5.32) 4 p j l i “ ´ i,j“1 l“1 } p q } ÿ ÿ Proof. This a consequence of the deﬁnition of scalar curvature and of Corollary A.5.4.

Proposition A.5.6. Let M,g be a Riemannian manifold and let T kM equipped p q with the Sasaki-type metric g. Then, T kM, g has constant scalar curvature if and p q only if M is ﬂat.

Proof. This is a direct consequence of Corollary A.5.5.

Corollary A.5.7. Let M, T kM as above. Then, T kM has constant scalar curvature with respect to the metric g if and only if the scalar curvature is zero.

Proof. This corollary is a consequence of Propositions A.5.6 and A.5.2.

Corollary A.5.8. Let M, T kM as above. Then, T kM is Einstein with respect to the metric g if and only if it is ﬂat.

Proof. Again, this is a direct consequence of Proposition A.5.6. Bibliography

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