Appendix an Overview of Morse Theory

Home , Hilbert manifold, Morse homology, Morse theory

Appendix

Morse theory is a beautiful subject that sits between differential geometry, topology and calculus of variations. It was first developed by Morse [Mor25] in the middle of the 1920s and further extended, among many others, by Bott, Milnor, Palais, Smale, Gromoll and Meyer. The general philosophy of the theory is that the topology of a smooth manifold is intimately related to the number and “type” of critical points that a smooth function defined on it can have. In this brief appendix we would like to give an overview of the topic, from the classical point of view of Morse, but with the more recent extensions that allow the theory to deal with so-called degenerate functions on infinite-dimensional manifolds. A compre- hensive treatment of the subject can be found in the first chapter of the book of Chang [Cha93]. There is also another, more recent, approach to the theory that we are not going to touch on in this brief note. It is based on the so-called Morse complex. This approach was pioneered by Thom [Tho49] and, later, by Smale [Sma61] in his proof of the generalized Poincaré conjecture in dimensions greater than 4 (see the beautiful book of Milnor [Mil56] for an account of that stage of the theory). The definition of Morse complex appeared in 1982 in a paper by Witten [Wit82]. See the book of Schwarz [Sch93], the one of Banyaga and Hurtubise [BH04] or the survey of Abbondandolo and Majer [AM06] for a modern treatment. We will try to keep our exposition as elementary as possible. In order to do this and avoid subtle technicalities, we will often not give the results in their maximal generality. Nevertheless, keeping in mind the application of Morse theory to the study of the critical points of the Lagrangian action functional, we will stress the regularity that the function under consideration must have, which will be mostly C1 and occasionally C2.

M. Mazzucchelli, Critical Point The ory for Lagrangian Systems, Progress in Mathematics 293, 157 DOI 10.1007/978-3-0348-0163-8, © Springer Basel AG 2012 158 Appendix: An Overview of Morse Theory

A.1 Preliminaries

Throughout this appendix we will denote by M a Hilbert manifold, i.e., a Haus- dorff topological space that is locally homeomorphic to a real separable Hilbert space E with smooth change of charts. If E is finite-dimensional, i.e., E = Rm for some m ∈ N,thenM is just an ordinary m-dimensional smooth manifold. In the context of Morse theory, the most relevant difference between the finite- dimensional and the infinite-dimensional situations is that in this latter case the manifold M is not locally compact. We will come back to this point later on. Throughout this appendix, F : M → R will be a C1 function, unless more regularity will be explicitly required. A point p ∈ M is called a critical point of F when the differential dF(p) vanishes. In this case, the corresponding image F(p) is called a critical value.WedenotebyCritF the set of critical points of F. In what follows, we will only deal with functions F having isolated critical points. We wish to investigate the relationship between the critical points of F and the topological properties (or, more precisely, the homological properties) of the underlying manifold M . Consider an open neighborhood U of a critical point p such that there exists −1 achartφ : U → E of M . We denote by Fφ the function F ◦ φ : E → R.By differentiating, we have

−1 dFφ(φ(p)) = dF(p) ◦ dφ (φ(p)).

2 Therefore φ(p) is a critical point of the function Fφ.IfF is C or at least twice Gâteaux-differentiable, then so is Fφ. We recall that, in these cases, the second Gâteaux derivative of Fφ at x can be seen as a symmetric bounded bilinear form HessFφ(x):E × E → R given by

HessFφ(x)[v, w] = (d(dFφ)(x)v) w, ∀v, w ∈ E.

In the above formula, we have denoted by d(dFφ)theGˆateaux derivative of dFφ, i.e., d d(dFφ)(x)v = dFφ(x + εv) ∀x, v ∈ E. dε ε=0

This latter expression coincides with the Fréchet derivative of dFφ in case F is C2. We define the Hessian of F at a critical point p as the symmetric bilinear form HessF(p):TpM × TpM → R given by HessF(p)[v, w]=HessFφ(x)[vφ, wφ], ∀v, w ∈ TpM , where x = φ(p), vφ =dφ(p)v and wφ =dφ(p)w are in E. It is easy to see that HessF(p) is intrinsically defined, i.e., its definition is independent of the chosen chart φ as long as p is a critical point of F. A.2. The generalized Morse Lemma 159

Remark A.1.1. If M is ﬁnite-dimensional and (x1,...,xm) is a system of local coordinates around p, the Hessian of F at the critical point p is given by

m ∂2F HessF(p)= (p)dxi ⊗ dxj. ∂xi ∂xj i,j=1

Moreover, if ∇ is any linear connection on M , the Hessian of F at p is given by

HessF(p)=∇(dF)(p).

We denote by ind(F,p) the supremum of the dimensions of those subspaces of TpM on which HessF(p) is negative-deﬁnite, and by nul(F,p) the dimension of the nullspace of HessF(p), i.e., the Hilbert space consisting of all v ∈ TpM such that HessF(p)[v, w] = 0 for all w ∈ TpM .Wecallind(F,p)andnul(F,p) respectively Morse index and nullity of the function F at p. Notice that both may be inﬁnite. Their sum is sometimes called the large Morse index.MorseTheory was initially developed for the so-called Morse functions, which are functions all of whose critical points have nullity equal to zero. Such critical points are called non-degenerate. Nowadays we are able to deal with functions having possibly degenerate critical points. Since the inner product · , · E of E is a non-degenerate bilinear form, there exists a bounded self-adjoint linear operator Hφ = Hφ(p):E → E such that

HessF(p)[v, w]= Hφvφ, wφ E , ∀v, w ∈ TpM .

By the spectral theorem, this operator induces an orthogonal splitting

0 − + E = Eφ ⊕ Eφ ⊕ Eφ ,

0 − + where Eφ is the kernel of Hφ,andEφ [resp. Eφ ] is a subspace of E on which Hφ is negative-deﬁnite [resp. positive-deﬁnite], i.e.,

0 Eφ = {v ∈ E | Hφv =0} , − Hφv, v E < 0, ∀v ∈ Eφ \{0} , + Hφw, w E > 0, ∀w ∈ Eφ \{0} .

− 0 Notice that ind(F,p)=dimEφ and nul(F,p)=dimEφ.

A.2 The generalized Morse Lemma

A starting point for Morse Theory might be the so-called Morse Lemma, originally due to Morse and generalized to inﬁnite-dimensional Hilbert manifolds by Palais [Pal63]. The version that we give here, valid for possibly degenerate functions, is due to Gromoll and Meyer [GM69a] and is sometimes referred to as the splitting 160 Appendix: An Overview of Morse Theory lemma. Since it is a local result, we can assume that our Hilbert manifold M is just the Hilbert space E.Let0 be an isolated critical point of a C2 function F : U → R,whereU is an open subset of E.WedenotebyH : E → E the bounded self- adjoint linear operator associated to the Hessian of F at 0,andbyE0 ⊕E− ⊕E+ of E the orthogonal splitting induced by the spectral theorem as in the previous section. According to this splitting, we will write every vector v ∈ E as

v = v0 + v±, where v± ∈ E± := E− ⊕ E+ and v0 ∈ E0. We assume that H is Fredholm, meaning that its image H(E) is closed and that its kernel and cokernel have both finite dimension. In particular nul(F, 0) is finite. Lemma A.2.1 (Generalized Morse Lemma). If V ⊆ U is a sufficiently small open neighborhood of 0, there exist • amapφ :(V , 0) → (U , 0) that is a homeomorphism onto its image, • a C2 function F 0 : V ∩ E0 → R whose Hessian vanishes at the origin, such that

F ◦ φ(v)=F 0(v0)+F ±(v±), ∀v = v0 + v± ∈ V , (A.1) where F ± : E± → R is the quadratic form deﬁned by

± ± 1 ± ± 1 ± ± ± ± F (v )= 2 HessF(0)[v , v ]= 2 Hv , v E, ∀v ∈ E .

Notice that 0 is a critical point of F ±, and therefore of F 0 as well. If 0 is a non-degenerate critical point of F, namely nul(F, 0) = 0, equation (A.1) reduces to

± 1 F ◦ φ(v)=F(0)+F (v)=F(0)+ Hv, v E . 2 This is a fundamental rigidity result for functions around a non-degenerate critical point: up to a local reparametrization and to an additive constant, they are all non-degenerate quadratic forms. In the general case, equation (A.1) gives us a local representation of a function F as the sum of a non-degenerate quadratic form F ± and of a function F 0 with a totally degenerate critical point at the origin.

A.3 Deformation of sublevels

At this point, let us recall some standard terminology from topology. Two continuous maps f0 and f1 from a topological space X to another topological space Y are homotopic, which we write as f0 ∼ f1, when there exists a continuous map f :[0, 1] × X → Y , called homotopy, such that f0 = f(0, ·)andf1 = f(1, ·). If we A.3. Deformation of sublevels 161 consider the two maps as maps of topological pairs f0,f1 :(X, A) → (Y,B), i.e., A ⊆ X, B ⊆ Y and f0(A)∪f1(A) ⊆ B, then they are homotopic (as maps of pairs) when there exists a homotopy f as before that further satisfies f(t, A) ⊆ B for all t ∈ [0, 1]. We write this as f :[0, 1] × (X, A) → (Y,B) and still f0 ∼ f1. A homotopy f is said to be relative to C when C is a subspace of X and f(t, x)=f(0,x) for all (t, x) ∈ [0, 1] × C. ∼ Amapj : X → Y is a homotopy equivalence, which we write as j : X −→ Y or X ∼ Y if j is implicit from the context, when there exists a map l : Y → X, called homotopy inverse, such that l◦j ∼ idX and j◦l ∼ idY . A map of topological pairs j :(X, A) → (Y,B) is a homotopy equivalence (of pairs) when it has a homotopy inverse of the form l :(Y,B) → (X, A) and the homotopies of j ◦ l and l ◦ j to the identities are homotopies of maps of pairs. Now, consider an inclusion ι : X→ Y and a retraction r : Y → X, i.e., a surjective map r such that the restriction r|X is equal to the identity idX .If there exists a homotopy R :[0, 1] × (Y,X) → (Y,X) such that R(0, ·)=idY and R(1, ·)=r,thenwesaythatY deformation retracts onto X and we call the homotopy R a deformation retraction.NoticethatR is assumed to be a homotopy of maps of pairs, therefore R(t, X) ⊆ X for all t ∈ [0, 1]. This implies that the inclusion ι is a homotopy equivalence. The deformation retraction R is called strong if we further assume that it is relative to X, i.e., R(t, x)=x for all (t, x) ∈ [0, 1] × X. After this excursion, let us go back to Morse Theory. So far, we have just discussed local aspects, but Morse Theory allows us to say something about the global properties of M and F : M → R. To start with, notice that the function F defines a filtration of the manifold M .Infact,ifc ∈ R, let us denote by (F)c the −1 open subspace F (−∞,c), called c-sublevel of F.If{cn | n ∈ N} is a monotone increasing sequence of real numbers tending to infinity, we have the sequence of inclusions

(F)c1 ⊆ (F)c2 ⊆ (F)c3 ⊆···⊆M . (A.2)

We also define (F)−∞ = ∅ and (F)+∞ = M . We wish to investigate the relation between the homology of pairs of sublevels ((F)b, (F)a) and the critical points of F contained in the region F −1[a, b), at first in the simple case in which the interval [a, b) contains a single critical value. Then, by standard algebraic- topological manipulations, we will derive some information about the homology of pairs ((F)b, (F)a), where the interval [a, b) contains an arbitrary number of critical values. Example A.1. From now on, it will be useful to keep in mind a classical finite- dimensional example: a torus M = T2 in R3 sitting on a plane as shown in Figure A.1. On this torus we consider the height function F : M → R, i.e., F(p) is the height of p ∈ T2 above the plane. For simplicity, we will assume that F is a Morse function. Its critical points p1, p2, p3 and p4 have indices 0, 1, 1 and 2 respectively. 162 Appendix: An Overview of Morse Theory

F p4

0 p1

3 Figure A.1. Height function on the 2-torus in R . The shaded region is the sublevel (F)c,for F(p2)

At this point, let us consider a Hilbert-Riemannian metric · , · · on M , i.e., a bounded bilinear form on TM that is symmetric and positive-deﬁnite, meaning

v, w q = w, v q , z,z q > 0 ∀q ∈ M , v,w,z ∈ TqM ,z=0 .

In local coordinates given by a chart φ : U → E, this metric can be expressed in terms of the inner product of E as

v, w q = Gφ(q)vφ, wφ E, ∀q ∈ M, v,w ∈ TqM , where Gφ is the unique map from U to the space of bounded self-adjoint operators on E that realizes this equality. If ψ : V → E is another chart of M ,then −1 ∗ −1 Gψ(q)= d(φ ◦ ψ )(ψ(q)) ◦ Gφ(q) ◦ d(φ ◦ ψ )(ψ(q)) ∀q ∈ U ∩ V .

We denote by ·· the associated Hilbert-Finsler metric given by vq = v, v q , ∀q ∈ M ,v∈ TqM .

This metric induces, as usual, a corresponding Hilbert-Finsler metric (still denoted ∗ by ··) on the cotangent bundle T M as

∗ νq =max{ν(v) | v ∈ TqM , vq =1} , ∀q ∈ M ,ν∈ TqM .

For each C1 curve σ :[a, b] → M , we deﬁne its length (with respect to the Hilbert-Finsler metric ··)as

b σ˙ (t)σ(t) dt. a A.3. Deformation of sublevels 163

If q and q are two points that belong to the same connected component of M ,we can define their distance (again, with respect to the Hilbert-Finsler metric ··) as the infimum of the lengths of all C1 curves joining q and q. If each component of M is a complete metric space with respect to this distance, we say that M is a complete Hilbert-Riemannian manifold. A C1 vector field X on M \ Crit(F) is called a pseudo-gradient of F when, for each q ∈ M , it satisfies

X (q)q ≤ 2dF(q)q, 2 dF(q) X (q) ≥dF(q)q.

By means of a partition of unity, one can show that pseudo-gradient vector ﬁelds always exist on Hilbert manifolds. Integrating X we obtain its (anti) pseudo- gradient ﬂow,whichisamap

ΦX : W → M , where W ⊆ R × M \ Crit(F)isanopenneighborhoodof{0}×M \ Crit(F), satisfying the following Cauchy problem:

∂ΦX (t, q)=−X (ΦX (t, q)), ∂t ΦX (0, ·)=idM .

It is easy to verify that the function F is decreasing along pseudo-gradient ﬂow lines. In fact, for every (t, q) ∈ W ,wehave

t F(ΦX (t, q)) − F(q)=− dF(ΦX (s, q)) X (ΦX (s, q)) ds 0 t 2 ≤− F X s, q s d (Φ ( )) ΦX (s,q) d 0 >0 < 0.

Example A.2. If F is C2, a pseudo-gradient is given by the gradient of F,that is the vector ﬁeld GradF deﬁned by

GradF(q),v q =dF(q)v, ∀q ∈ M ,v∈ TqM .

We would like to use the pseudo-gradient ﬂow to deform a certain sublevel

(F)c2 to a lower one, say (F)c1 ,forsomec1

Figure A.2. Deformation of (F)c2 over (F)c1 along gradient ﬂow lines in the torus Example A.1.

q1 q2 q3 q4 q

Figure A.3.

F was found in the 1960s by Palais and Smale [PS64], and it now carries their name: we say that F satisﬁes the Palais-Smale condition at level c when, for each sequence {qn | n ∈ N}⊆M such that

lim F(qn)=c, n→∞

lim dF(qn) =0, (A.3) n→∞ qn there exists a subsequence converging to a point q ∈ M . By (A.3), the limit point q must be a critical point of F.WesaythatF satisfies the Palais-Smale condition if it satisfies it at every level c ∈ R. Example A.3. The following are two slightly different examples of situations that the Palais-Smale condition wants to avoid. • Consider the function F(q)=exp(−q)onM = R. For any diverging sequence qn ↑∞we have that F(qn) → 0andF (qn) → 0, however {qn} does not admit any converging subsequence (see Figure A.3). • Consider a function F : M → R such that, for a certain level c ∈ R,theset CritF ∩ F −1(c) is not compact (e.g., F : R → R with F(q)=sin(q)and A.4. Passing a critical level 165

−1 c = 1). Then there is a sequence of critical points {qn}⊂F (c)thatdoes not admit any converging subsequence.

As promised, here is the important consequence of the Palais-Smale condition. Lemma A.3.1 (Deformation Lemma). Let M be a complete Hilbert-Riemannian manifold and assume that F : M → R satisﬁes the Palais-Smale condition at every level c ∈ [a, b] and does not have any critical value in the interval [a, b). Then the inclusion (F)a → (F)b is a homotopy equivalence.

Notice that, by this lemma, the inclusion (F)a → (F)b induces the homology isomorphism H∗((F)a) −→ H∗((F)b), and H∗((F)b, (F)a) = 0. The same is true if we substitute the singular homology with any other homotopy invariant functor: singular cohomology, K-theory, and so forth.

A.4 Passing a critical level

We now want to study the changes that occur, in term of homology, whenever we pass a critical level. In order to do this, we ﬁrst introduce a fundamental invariant of isolated critical points p of a C1 function F : M → R.Thelocal homology of F at p is deﬁned as the relative homology group

C∗(F,p):=H∗((F)c ∪{p} , (F)c), where c = F(p). This is a local invariant, in the sense that it depends only on the germ of F at p.Infact,ifU ⊆ M is an open neighborhood of p, by excision we obtain that the inclusion

(U ∩ (F)c ∪{p} , U ∩ (F)c) → ((F)c ∪{p} , (F)c) induces an isomorphism in homology, and therefore the local homology of F at p coincides with H∗(U ∩ (F)c ∪{p} , U ∩ (F)c).

Remark A.4.1. It is easy to see that the local homology of F at p can be equiva- lently deﬁned as H∗ (F)c, (F)c \{p} . The role that local homology plays in Morse Theory is illustrated by the following statements. We recall that a λ-cell,forλ ∈ N,issimplyaλ-dimensional closed disk Dλ. A topological space Y is obtained by the attachment of a λ-cell to a topological manifold X when there exists a continuous map f : ∂Dλ → ∂X such that Y = X ∪ Dλ/ ∼, where ∼ is the identiﬁcation given by z ∼ f(z)foreachz ∈ ∂Dλ. 166 Appendix: An Overview of Morse Theory

c2 p2

F,p2 F Figure A.4. Attachment of a cell of dimension 1 = ind( ) to the closed sublevel ( )c1 of the height function in Example A.1. Notice that the result of the attachment is F homotopically equivalent to ( )c2 .

Theorem A.4.1. Consider a complete Hilbert-Riemannian manifold M ,aC2 function F : M → R that satisﬁes the Palais-Smale condition, and an interval (c1,c2) ⊂ R. Assume that p is the only critical point of F with critical value c1,c2 F inside ( ), and that it is non-degenerate. Then ( )c2 is homotopically equiv- F F,p alent to ( )c1 with an ind( )-cell attached. In particular

F n =ind(F,p), Cn(F,p)= 0 n =ind( F,p), where F is the coeﬃcient group of the homology.

In Figure A.4 the assertions are illustrated for the case of the height function on the torus of Example A.1. Notice that Theorem A.4.1 implies that critical points p with infinite Morse index do not produce any change in the homotopy type as we cross their sublevel (in particular their local homology C∗(F,p)istrivial).This follows from the fact that the unit sphere of an infinite-dimensional Hilbert space is contractible, as one can easily prove by constructing an explicit deformation retraction to a point (a stronger and much harder theorem due to Bessaga [Bes66] asserts that every infinite-dimensional Hilbert space is even diffeomorphic to its unit sphere). Theorem A.4.1 tells us that, for a non-degenerate critical point of a C2 function, the knowledge of the local homology at it coincides with the knowledge of its Morse index. This is no longer true in the degenerate case, in which the Morse index and nullity of a critical point do not completely determine its local homology. An easy example on M = R2 is the following. Example A.4. Consider the functions F, G : R2 → R given by

F(x, y)=(y − 2x2)(y − x2), G (x, y)=x4 + y2. A.4. Passing a critical level 167

0 x

Figure A.5. Behaviour of F(x, y)=(y − 2x2)(y − x2) around the critical point 0.Theshaded −1 region corresponds to the sublevel (F)0 = F (−∞, 0).

Both functions have 0 =(0, 0) as isolated critical point with

ind(F, 0)=ind(G , 0)=0, nul(F, 0)=nul(G , 0)=1.

However 0 is a saddle point for F (see Figure A.5) and a global minimum for G , therefore

F n =1, F n =0, Cn(F, 0)= Cn(G , 0)= 0 n =1 , 0 n =0 .

If the function F is merely of class C1 and has possibly degenerate critical points, we can still describe the homological change that occurs by crossing a critical level in terms of local homology. Theorem A.4.2. Consider a complete Hilbert-Riemannian manifold M and a C1 function F : M → R satisfying the Palais-Smale condition. Assume that the critical points of F having critical value c are isolated (hence there are only ﬁnitely many),andchooseε>0 small enough such that F does not have critical values in (c, c + ε). The following claims hold: (i) For each p ∈ Crit(F) with F(p)=c, the inclusion

jp :((F)c ∪{p} , (F)c) → ((F)c+ε, (F)c)

induces the homology monomorphism

H∗(jp):C∗(F,p) → H∗((F)c+ε, (F)c).

(ii) If {p1,...,ps}⊂M is the set of critical point of F with critical value c,we have an isomorphism

Jc :C∗(F,p1) ⊕···⊕C∗(F,ps) −→ H∗((F)c+ε, (F)c),

where Jc =H∗(jp1 ) ⊕···⊕H∗(jps ). 168 Appendix: An Overview of Morse Theory

By playing with the ﬁltration (A.2) given by the sublevels of F and with the associated long exact sequences in homology we can deduce the following statement. Theorem A.4.3. Consider a complete Hilbert-Riemannian manifold M and a C1 function F : M → R that satisﬁes the Palais-Smale condition and has isolated critical points. Fix a bounded interval [a, b] ⊂ R,andlet{c1,...,ct} be the set of critical values of F inside [a, b).Thenforeachε such that

0 <ε

Moreover ∞ ∞ t − n F , F − n F , F , ( 1) rank Hn(( )b ( )a)= ( 1) rank Hn(( )ch+ε ( )ch ) n=0 n=0 h=1 provided the above sums are ﬁnite. As a consequence of this result and of Theorem A.4.2, we obtain the cele- brated Morse inequalities. Corollary A.4.4 (Morse inequalities). Assume that F satisﬁes the hypotheses of Theorem A.4.3. For each bounded interval [a, b] ⊂ R,if{p1,...,pu}⊂M is the set of critical points of F with critical values inside [a, b),wehave u rank Hn((F)b, (F)a) ≤ rank Cn(F,ph). ∀n ∈ N (A.4) h=1

A.5 Local homology and Gromoll-Meyer pairs

In their work on degenerate Morse theory [GM69a], Gromoll and Meyer showed that the local homology of an isolated critical point can also be computed as the homology of a suitable closed neighborhood of the critical point relative to a part of its boundary. The homotopy type of the pair (closed neighborhood, part of boundary) is what in the 1980s, after the seminal work of Conley [Con78], would be called Conley index of the critical point. In Conley theory, the critical point is viewed as an isolated invariant set for the dynamical system deﬁned by a pseudo-gradient A.5. Local homology and Gromoll-Meyer pairs 169

ﬂow. We refer the interested reader to Chang and Ghoussoub [CG96] for a detailed investigation of the relation between Gromoll-Meyer theory and Conley theory. For this section, let us assume that our function F : M → R is C2 with Fredholm Hessian, so that we can choose as a pseudo-gradient X the gradient of F, see Example A.2. Some, but not all, of the statements that we will give still hold under the C1 assumption on F (up to choosing a C1 pseudo-gradient). Let p ∈ M be a critical point of F with F(p)=c. A pair of topological spaces (W , W−) is called a Gromoll-Meyer pair for F at p when (GM1) W ⊂ M is a closed neighborhood of p that does not contain other critical points of F, (GM2) there exists ε>0 such that [c − ε, c) does not contain critical values of F,andW ∩ (F)c−ε = ∅,

(GM3) if t1

(GM4) W− is deﬁned as W− = q ∈ W max{t ∈ R | ΦX (t, q) ∈ W } =0

and it is a piecewise submanifold of M transversal to the ﬂow ΦX . GivenanopenneighborhoodU of the isolated critical point p, it is always possible to build a Gromoll-Meyer pair (W , W−) such that W ⊂ U . Moreover, all the Gromoll-Meyer pairs of a critical point have the same homology type, and in fact they provide an alternative deﬁnition for the local homology.

Theorem A.5.1. If (W , W−) is a Gromoll-Meyer pair for F at an isolated critical point p ∈ M ,wehave H∗(W , W−) C∗(F,p). Now, let us apply the generalized Morse Lemma (Lemma A.2.1) and the notation adopted therein: without loss of generality, we can identify an isolated critical point p of F with the origin 0 in the Hilbert space E, and we can assume that F is deﬁned on a neighborhood V ⊂ E of 0 and has the form

F(v)=F 0(v0)+F ±(v±), ∀v = v0 + v± ∈ V ⊂ E = E0 ⊕ E±.

The origin is a non-degenerate critical point of F ± : V ∩ E± → R and a totally degenerate critical point of F 0 : V ∩E0 → R. If we consider Gromoll-Meyer pairs ± ± 0 0 ± 0 (W , W− )and(W , W− )forF and F respectively at 0, it is easy to verify that the product of these pairs, that is

± ± 0 0 ± 0 ± 0 ± 0 (W , W− ) × (W , W− )=(W × W , (W− × W ) ∪ (W × W− )), is a Gromoll-Meyer pair for F at 0. Now, assume that the coefficient group F of the homology is a field. By the Künneth formula we get an isomorphism of graded 170 Appendix: An Overview of Morse Theory vector spaces ± ± 0 0 ± ± 0 0 H∗((W , W− ) × (W , W− )) H∗(W , W− ) ⊗ H∗(W , W−), and by the above Theorem A.5.1 we obtain the following. ± 0 Theorem A.5.2. C∗(F, 0) C∗(F , 0) ⊗ C∗(F , 0). By Theorem A.4.1, the local homology of the non-degenerate function F ± at 0 is nontrivial only in degree ind(F ±, 0), where it coincides with the coefficient vector space F. Notice that the Morse index of F ± at 0 is precisely the Morse index of F at 0, and therefore Theorem A.5.2 readily gives the following fundamental result. 0 Theorem A.5.3 (Shifting). C∗(F, 0) C∗−ind(F,0)(F , 0).

Corollary A.5.4. The local homology group Cn(F, 0) is trivial if nind(F, 0)+nul(F, 0).

A.6 Minimax On a ﬁnite-dimensional closed manifold, an elementary way to detect a critical value of a given function is to look for its global minima (or maxima). A more general class of critical values, that we are going to discuss in this section, is given by the so-called “minimax” values. Let M be a complete Hilbert-Riemannian manifold, F : M → R a C1 function and U a family of subsets of M . We deﬁne the minimax of F over the family U as minimax F := inf sup {F(p)}∈R ∪{±∞}. U U ∈U p∈U

The following statement guarantees that, under certain conditions, the minimax is a critical value of F. We refer the reader to [HZ94, page 79] for a proof. Theorem A.6.1 (Minimax Theorem). Assume that the following conditions are satisfied: (i) F satisfies the Palais-Smale condition, (ii) there exists a pseudo-gradient X for F whose anti pseudo-gradient flow is defined for all the non-negative times, i.e.,

ΦX :[0, ∞) × M \ Crit(F) → M ,

(iii) the family U is positively invariant under the anti pseudo-gradient ﬂow of X , i.e., for each U ∈ U and t ≥ 0 we have ΦX (t, U ) ∈ U, (iv) minimax F is ﬁnite. U Then minimax F is a critical value of F. U A.6. Minimax 171

Remark A.6.1. If we take U to be the family of singletons on M , i.e.,

U = {{p}|p ∈ M }, then condition (iii) above is trivially satisﬁed and

minimax F =min{F(p)} . U p∈M

Notice that condition (ii) above is automatically satisﬁed for any pseudo- gradient if the function F is bounded from below. As for the choice of a suitable family U satisfying conditions (iii) and (iv), there are several possibilities. In the following we discuss one of these, which leads to a homological version of the minimax theorem. Let a be a nonzero homology class in Hd(M ). We denote by Sa the family consisting of the supports of the singular cycles in M representing a, i.e., V V S α d α α , α a , a = v(Δ ) = v [ ]= v=1 v=1

d where each αv is a singular simplex of the cycle α, i.e., αv :Δ → M .Notice that, by the homotopic invariance of singular homology, Sa satisfies condition (iii) of the minimax theorem. Moreover, Sa is a family of compact subsets of M . Therefore, if the function F is bounded from below, its minimax over the family Sa is finite. Hence, we have the following. Theorem A.6.2 (Homological Minimax Theorem). Let M be a complete Hilbert- Riemannian manifold, and F : M → R a C1 function that is bounded from below and satisfies the Palais-Smale condition. Then for each nonzero a ∈ H∗(M ) the quantity minimaxSa F is finite and it is a critical value of F. If F is also C2, then by the following result due to Viterbo [Vit88] it is sometimes possible to estimate the Morse index and nullity pair of a critical point corresponding to the minimax critical value. Theorem A.6.3. Under the assumptions of Theorem A.6.2,ifF is also C2 and its critical points corresponding to the critical value minimaxSa F are isolated, then there exists a critical point p of F such that

F(p) = minimax F, Sa ind(F,p) ≤ d ≤ ind(F,p)+nul(F,p), where d is the degree of a, i.e., a ∈ Hd(M ). Now, if two linearly independent homology classes are given, one might ask whether or not their associated minimax values (with respect to the function F) are the same. Before answering this question, let us quickly recall some deﬁnitions 172 Appendix: An Overview of Morse Theory from algebraic topology. Given a topological space X and two nonzero homology d classes a, b ∈ H∗(X), we write a ≺ b when there exists ω ∈ H (X), for some d>0, j such that a = b ω. Here, :Hi(X) ⊗ H (X) → Hi−j (X)isthecap product, which is related to the cup product :Hi(X) ⊗ Hj(X) → Hi+j (X)by

∗ (ωψ)(a)=ψ(a ω), ∀ω, ψ ∈ H (X),a∈ H∗(X). (A.5)

The cup product gives an important homotopical invariant of X,thecup-length CL(X), deﬁned as the maximum integer l ≥ 0 such that there exist cohomology d classes ω1,...,ωl, with ωi ∈ H i (X)anddi > 0foreachi =1,...,l, satisfying

ω1 ···ωl =0 .

The relation “≺” defined above can be used to give an alternative definition of the ∗ cup-length. In fact, if ω1 ···ωl =0inH (X), then there exists al ∈ H∗(X) such that (ω1 ···ωl)(al) = 0, and we can iteratively define ai−1 := ai ωi. Therefore, by (A.5), we have

ω1(a1)=(ω2 ω1)(a2)=···=(ωl ···ω1)(al) =0 , (A.6) which implies that a0 = a1 ω1 is nonzero and, by construction, a0 ≺ ··· ≺ al. d Conversely, if a0 ≺ ··· ≺ al, then there exists a nonzero ωi ∈ H i (X) such that di > 0andai−1 = ai ωi for each i =1,...,n, so that (A.6) holds and we have ∗ that ω1 ···ωl =0inH (X). Summing up, we have obtained the following. Proposition A.6.4. The cup-length CL(X) is equal to the maximum l ∈ N ∪{0} such that there exist l +1nonzero homology classes a0,...,al ∈ H∗(X) satisfying a0 ≺···≺al. Coming back to the question as to when two given homology classes have different minimax values with respect to some function, we have the following result. Theorem A.6.5. Let M be a complete Hilbert-Riemannian manifold, and F : M → R a C1 function that is bounded from below and satisfies the Palais-Smale condition. Consider non-zero homology classes a, b ∈ H∗(M ) such that a ≺ b. If F has only finitely many critical points corresponding to the critical value F minimaxSb ,then minimax F < minimax F. Sa Sb This theorem, together with Proposition A.6.4, readily implies the following multiplicity result for the critical points of F. Corollary A.6.6. Under the assumptions of Theorem A.6.5, the function F has at least CL(M )+1critical points. Bibliography

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Chapter 1 M Smoothfinite-dimensionalclosedmanifold ...... 2 TM Tangent bundle of M ...... 2 L Lagrangianfunction ...... 2 t0,t1 A Action functional for curves parametrized on [t0,t1] ...... 2 U Atlas for M ...... 2 (Uα,φα)ChartinU ...... 2 TU Atlas for TM associated to U ...... 2 (TUα, Tφα)ChartinTU ...... 2 T∗U Atlas for T∗M associated to U ...... 2 ∗ ∗ ∗ (T Uα, T φα)ChartinTU ...... 2 ∂vL Fiberwise derivative of L ...... 2 γ∗TM Pull-back of TM by γ ...... 3 XL Euler-Lagrange vector field of L ...... 4 ΦL Euler-Lagrange flow of L ...... 5 T∗M Cotangent bundle of M ...... 5 H Hamiltonianfunction ...... 5 λ Liouville form on T∗M ...... 5 ω Canonical symplectic form on T∗M ...... 5 XH Hamiltonian vector field of H ...... 6 Interiorproduct ...... 6 ΦH Hamiltonian flow of H ...... 6

LegL Legendre transform given by L ...... 6

·, ·· Riemannianmetric ...... 7 dist Distance induced by the Riemannian metric on M ...... 7 L Convex function on Rm ...... 8 H Legendre-dual of L ...... 9 μLeb Lebesguemeasure ...... 12 Leb(V) Lebesgue number of the open cover V ...... 20

179 180 List of Symbols

Chapter 2 A Actionfunctionalfor1-periodiccurves ...... 29 Bγ Second variation of A at γ ...... 30 E RealseparableHilbertspace ...... 31 Bil(E) Banach space of bounded bilinear forms on E ...... 31 B Bounded symmetric bilinear form on E ...... 31 B Bounded self-adjoint linear operator on E ...... 31 ind(B) Morse index of B ...... 31 nul(B) Nullity of B ...... 31 ind(γ) = ind(A ,γ) Morse index of A at γ ...... 33 nul(γ)=nul(A ,γ) Nullity of A at γ ...... 33 γ[n] nth iteration of γ ...... 35 A [n] Action functional for n-periodiccurves ...... 36 [n] B Self-adjoint bounded operator associated to Bγ[n] ...... 36 E[n] Space of W 1,2 sections of γ∗TM ...... 36 [n] 1,2 Ez Space of (n, z)-periodic W complex sections of γ∗TM ...... 36 [n] [n] indz,λ(γ )(z,λ)-Morse index of γ ...... 37 [n] [n] indz(γ ) z-Morse index of γ ...... 37 [n] [n] nulz(γ ) z-nullity of γ ...... 37 ind(γ) Mean Morse index of γ ...... 40 2m ω0 Standard symplectic form of R ...... 42 2m J0 Standard complex structure of R ...... 42 Sp(2m) Symplectic group of R2m ...... 42 GL(2m) General linear group of R2m ...... 42 Sp0(2m) Automorphisms in Sp(2m) witheigenvalue1 ...... 42 Sp∗(2m) Complementary of Sp0(2m)inSp(2m) ...... 42 Sp±(2m) Connected components of Sp∗(2m) ...... 42 O(2m) Orthogonal group of R2m ...... 42 r Retraction of Sp(2m) onto Sp(2m) ∩ O(2m) ...... 42 GL(m, C) General linear group of Cm ...... 43 U(m) Unitary group of Cm ...... 43 detC Complexdeterminant ...... 43 ρ Rotationfunction ...... 43 P Continuous paths in Sp(2m) fromI ...... 43 P∗ Paths of P with endpoint in Sp∗(2m) ...... 43 P0 Paths of P with endpoint in Sp0(2m) ...... 43 Ψ • Ψ˜ Concatenation of the paths Ψ and Ψ˜ ...... 43 ι(Ψ) MaslovindexofΨ ...... 43 ν(Ψ) Multiplicityoftheeigenvalue1 ofΨ(1) ...... 44 Γ Hamiltonianperiodicorbit ...... 45 Vm Vertical Lagrangian subspace of R2m ...... 46 Γφ Path in Sp(2m) deﬁned by Γ and φ ...... 46 List of Symbols 181

Sp(2m, Vm) Symplectic automorphisms preserving Vm ...... 46 π1 Fundamental group functor ...... 47 (ι(Γ),ν(Γ)) MaslovindicesoftheperiodicorbitΓ ...... 48

Chapter 3 I Interval [0, 1] ...... 50 C0(I; M) C0 free path space of M ...... 50 C∞(I; M) C∞ free path space of M ...... 50 W 1,2(I; M) W 1,2 free path space of M ...... 50 W 1,2(I; R2m+1) Sobolevspace ...... 50 W 1,2(γ∗TM) W 1,2 sections of γ∗TM ...... 50 ∇t Covariantderivative ...... 50 1,2 ∗ · , ·γ Inner product on W (γ TM) ...... 50 exp Exponential map of (M,·, ··) ...... 51 −1 1,2 (Uγ , expγ )ChartofW (T; M) ...... 51 1,2 1,2 WV (I; M) Submanifold of W (I; M) ...... 51 1,2 ∗ 1,2 ∗ WV (γ TM) Hilbert subspace of W (γ TM) ...... 52 W 1,2 I M W 1,2 q q q0,q1 ( ; ) paths joining 0 and 1 ...... 52 1,2 ∗ 1,2 ∗ W0 (γ TM) Hilbert subspace of W (γ TM) ...... 52 T Circle R/Z ...... 52 W 1,2(T; M) W 1,2 free loop space of M ...... 52 1,2 (Uγ , Φγ )ChartofW (I; M) ...... 53 1,2 m 1,2 m W0 (I; R ) Hilbert subspace of W (I; R ) ...... 53 1,2 m 1,2 m WV (I; R ) Hilbert subspace of W (I; R ) ...... 53 T[n] Circle R/nZ ...... 53 ψ[n] nth iterationmap ...... 54 {hε | ε>0} Approximateidentity ...... 55 ∗ Convolutionoperator ...... 55 Top Categoryoftopologicalspacesandcontinuousmaps ...... 58 C Category ...... 58 F Functor between Top and C ...... 58 1,2 1,2 Wcontr(T; M) Contractible loops in W (T; M) ...... 58 ev Evaluationmap ...... 58 Grp Categoryofgroupsandhomomorphisms ...... 59 Bil(W 1,2(I; Rm)) Bounded bilinear forms on W 1,2(I; Rm) ...... 67 A [n] Mean action functional on W 1,2(T[n]; M) ...... 77

Chapter 4

t0,t1 A Action functional for curves parametrized on [t0,t1] ...... 80

γq0,q1 Unique action minimizer with endpoints q0 and q1 ...... 81 B(q,R) Riemannian closed ball with center q and radius R ...... 81 Δ(ρ0) ρ0-neighborhood of the diagonal of M × M ...... 84 182 List of Symbols

QL Projection of the Euler-Lagrange flow onto M ...... 85 Zk Cyclic group of order k ...... 86 ∞ Ck (T; M) Space of continuous and k-brokensmoothloops ...... 86 Δk Neighborhood of the diagonal of the k-fold product of M ...... 87 ∞ λk Embedding of Δk into Ck (T; M) ...... 87 Λk k-brokenEuler-Lagrangeloopspace ...... 87 γq Broken Euler-Lagrange loop between the k-uple of points q ...... 87 Ak Discreteactionfunctional ...... 89 L QuadraticLagrangian ...... 93 A Action functional associated to L ...... 93 E RealHilbertspace ...... 96 B Bounded symmetric bilinear form on E ...... 96 ind(B) Morse index of B ...... 96 {En | n ∈ N} Family of Hilbert subspaces of E ...... 96 Pn Orthogonal projector of E onto En ...... 96 V Finite-dimensional subspace of E ...... 96 S(V ) Sphere of V ...... 96 m m Affk(T; R ) k-broken affine loop space of R ...... 97 · Integerpart ...... 97 (A )c c-open sublevel of A ...... 100 (Ak)c c-open sublevel of Ak ...... 100 rk Retraction of a sublevel of A onto a sublevel of Ak ...... 100 Rk Deformation retraction from the identity to rk ...... 100 H∗ Singularhomologyfunctor ...... 102 C∗(Ak, q) Local homology of Ak at q ...... 102 C∗(A ,γq ) Local homology of A at γq ...... 102 H∗(λk) Homology homomorphism induced by λk ...... 102 CL(W 1,2(T; M)) Cup-length of the free loop space W 1,2(T; M) ...... 103 [n] ∞ [n] λk Embedding of Δnk into Ck (T ; M) ...... 106 [n] Λk k-broken n-periodicEuler-Lagrangeloopspace ...... 106 [n] Ak Discretemeanactionfunctional ...... 107 [n] ψk Discrete nth-iterationmap ...... 107 q[n] nth-iteration of q ...... 107

Chapter 5 E RealseparableHilbertspace ...... 110 U Open set in E ...... 110 F Function over U ...... 110 H Operator on E associated to the Hessian of F at x ...... 110 E• Hilbert subspace of E ...... 110 List of Symbols 183

J Inclusion map of U ∩ E• into U ...... 110 GradF Gradient of F ...... 110 C∗(F , x) Local homology of F at x ...... 110 F• Restriction of F to E• ...... 110 C∗(F•, x) Local homology of F• at x ...... 110 H∗(J) Local homology homomorphism induced by J ...... 110 ind(F , x) Morse index of F at x ...... 111 nul(F , x) Nullity of F at x ...... 111 ind(F•, x) Morse index of F• at x ...... 111 nul(F•, x) Nullity of F• at x ...... 111 E+ ⊕ E− ⊕ E0 Orthogonal splitting of E ...... 112 F ± + F 0 Decomposition of F ...... 113 H• Operator on E• associated to the Hessian of F• at x ..... 117 + − 0 E• ⊕ E• ⊕ E• Orthogonal splitting of E• ...... 117 ± 0 F• + F• Decomposition of F• ...... 117

Chapter 6 A TonelliActionfunctional ...... 129 A [n] MeanTonelliActionfunctional ...... 129 LR R-modiﬁcation of L ...... 129 2 ∂vv FiberwiseHessianoperator ...... 129 AR Action functional of LR ...... 131 [n] AR Mean Action functional of LR ...... 131 Kβ Support of the cycle β ...... 134 Ak DiscreteTonelliaction ...... 136 [n] Ak DiscretemeanTonelliaction ...... 138 Δp Standard p-simplex in Rp ...... 140 [n] Banσ Bangerthomotopy ...... 140 α Inverse of the path α ...... 141 α • β Associative composition of the paths α and β ...... 141 ev Evaluationmap ...... 142 ϑx ,ϑx0 x0 x Horizontalbrokengeodesics ...... 142 ϑn Path obtained from ϑ bytheBangertconstruction ...... 142 ϑ Path of pulling loops ...... 142 v1,...,vj Convex hull of v1,...,vj ...... 146 Kp Set of non-negative integer powers of p ...... 152

Appendix E RealseparableHilbertspace ...... 158 M Hilbertmanifold ...... 158 U Opensetina Hilbertmanifold ...... 158 F Functionovera Hilbertmanifold ...... 158 184 List of Symbols

CritF Set of critical points of F ...... 158 Fφ Pull-back of F via the chart φ ...... 158 HessFφ Hessian of Fφ ...... 158 HessF Hessian of F ...... 158 vφ Representation of v ∈ TpM via the chart φ ...... 158 Hφ Operator on E associated to the Hessian of Fφ at φ(p) . . . 159 + − 0 Eφ ⊕ Eφ ⊕ Eφ Orthogonal splitting of E ...... 159 F ± + F 0 Decomposition of F ...... 160 f0 ∼ f1 Homotopicmaps ...... 160 ∼ j : X −→ Y Homotopyequivalence ...... 161 X ∼ Y Homotopyequivalentspaces ...... 161 −1 (F )c Sublevel F (−∞,c) ...... 161 X Pseudo-gradientvectorfield ...... 163 ΦX Anti pseudo-gradient flow of F ...... 163 GradF Gradient of F ...... 163 H∗ Singularhomologyfunctor ...... 165 C∗(F ,p) Local homology of F at p ...... 165 Dλ Closed disk of dimension λ ...... 165 F Coefficientgroupforsingularhomology ...... 166 (W , W−) Gromoll-Meyerpair ...... 169 U Family of subsets of M ...... 170 minimax F Minimax of F over U ...... 170 U Sa Family of the supports of those cycles representing a ...... 171 ≺ Homological relation induced by the cap product ...... 172 Capproduct ...... 172 Cupproduct ...... 172 CL(X) Cup-length of X ...... 172 Index

absolutely continuous curve, 11 Conley conjecture, 151 absolutely equicontinuous space, 15 Conley index, 168 action functional Conley-Zehnder index, 44 Hamiltonian –, 7 conservation of energy, 25 mean, 152 convex hull, 146 Lagrangian –, 2, 29 convex quadratic modification, 129 discrete –, 89 critical point, 158 discrete mean–, 107 non-degenerate, 159 discrete Tonelli –, 136 totally degenerate, 160 mean –, 77 cup product, 172 second variation, 30 cup-length, 103, 134, 172 action minimizer, 11 cyclic group, 86 unique, 16 almost complex structure Euler-Lagrange compatible, 45 flow, 5 2m standard on R ,42 global, 5, 25 approximate identity, 55 system, 4, 70 assumptions vector field, 4 (GM1)–(GM4), 169 evaluation map, 58, 142 (M1)–(M2), 129 extremal, 2 (Q1)–(Q2), 59 (T1)–(T2), 7 Fenchel relation, 9 fiber product, 87 Bangert homotopy, 140, 144 fiberwise derivative, 2 boot-strap, 72 filtration, 161 bounded bilinear form, 31, 67 Fourier expansion, 38 broken Fredholm Euler-Lagrange loop, 87 operator, 31, 160 n-periodic, 106 spectrum of –, 31 smooth loop, 86 symmetric bilinear form, 31 semipositive-definite, 32 cap product, 172 free loop space cell, 165 C0(T; M), 54 complex determinant, 43 C∞(T; M), 54 ∞ configuration space, 2 Ck (T; M), 86 m conjugacy classes of π1(M), 56, 77, 133 Affk(T; R ), 97

185 186 Index

Λk,87 Jacobi W 1,2(T; M), 52 system, 94 1,2 Wcontr(T; M), 58 vector field, 94 connected components, 56, 77 free path space, 50 Künneth formula, 122, 169 C0(I; M), 50 kinetic tensor, 63 C∞(I; M), 50 1,2 W (I; M), 50 Lagrangian function, 2 geodesic autonomous, 2, 25 horizontal –, 142 convex quadratic-growth, 60 horizontal broken –, 142 electro-magnetic, 63 geometrically distinct, 151 modification of –, 129 gradient, 163 non-degenerate, 4 Gromoll-Meyer pair, 169 Tonelli, 7 subspace, 46 Lebesgue number, 20, 81 H-space, 104 Legendre Hamilton-Jacobi equation, 16 dual function, 6, 9 Hamiltonian transform, 6 flow, 6 lemma function, 5 Bangert homotopy –, 140 autonomous, 5, 25 deformation, 165 coercive, 25 Du Bois-Reymond –, 72 Legendre-dual, 6 fund. of calculus of variations, 3 Tonelli, 10 generalized Morse –, 160 system, 6 revisited, 113 vector field, 6 Gronwall –, 27 Hessian, 158 splitting –, 160 fiberwise, 129 Levi-Civita connection, 50 Hilbert manifold, 158 Lie group, 104 homological vanishing, 139 Liouville form, 5 homotopical vanishing, 140 local homology, 165 homotopy, 160 of the action functional, 102 inverse, 161 localization, 63, 87 Bangert –, 140, 144 equivalence, 161 magnetic form, 63 relative to a subspace, 161 manifold Fréchet –, 86 injectivity radius, 50, 142 fiber product of –, 87 iteration, 35 Hilbert –, 50 inequalities, 40 Hilbert-Riemannian –, 162 map, 54 complete –, 163 discrete –, 107 Maslov index, 48 Index 187 metric Fredholm, 31 Hilbert-Finsler –, 162 semipositive-definite –, 32 Hilbert-Riemannian –, 162 symplectic minimax, 103, 170 diffeomorphism, 45 Morse group, 42 complex, 157 universal cover, 44 function, 159 structure, 5 ∗ index, 31, 159 canonical on T M,5 2m large –, 31, 159 standard on R ,42 mean –, 40 trivialization, 46 inequalities, 156, 168 lemma, 160 theorem revisited, 113 Arzelà-Ascoli –, 15, 35, 74 nullity, 31, 159 Cauchy-Lipschitz –, 4, 6 theory, 157 Conley conjecture, 151 existence of modifications, 129 Palais-Smale condition, 73, 164 index –, 48 path space minimax, 170 W 1,2 I M homological, 171 q0,q1 ( ; ), 52 periodic orbit regularity of minimizers, 24 contractible, 45 Rellich-Kondrachov –, 34 homologically visible, 152 shifting –, 170 strongly –, 152 smooth dependence on endpoints, 84 phase space, 5 Sobolev embedding –, 50, 54 polar decomposition, 42 Tonelli –, 11 potential energy, 63 convex quadratic-growth case, 76 uniqueness of minimizers, 81 pseudo-gradient, 163 vanishing flow, 163 homological –, 139 pulling loop, 142 homotopical –, 140 Weierstrass –, 16 retraction, 161 Whitney embedding –, 50 deformation –, 161 strong–, 161 uniformly integrable space, 12 Riemann tensor, 94 unitary group, 43 rotation function, 43 variation of a curve, 2 simple function, 13 singular simplex, 140 Sobolev space, 50 stationary curve, 17 sublevel, 161 action –, 82, 90 support of a singular cycle, 171 symmetric bilinear form compact –, 32