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Riemannian Geometry Appendix A Riemannian Geometry In this appendix besides introducing notation and basic definitions we recollect some of the more technical results in Riemannian geometry we exploited in these lecture notes. We freely refer to the excellent exposition of the theory provided by [4, 8], and to [27] for further details. A.1 Notation In what follows, Vn will always denote a C1 compact n–dimensional manifold, (n 3). Unless otherwise stated, the manifold Vn will be supposed oriented, connected, and without boundary. Let .E; Vn;/be a smooth vector bundle over Vn with projection map W E ! Vn. When no confusion arises the space of smooth 1. n; / n 1. n; / sections fs 2 C V E j ı s D idV g will simply be denoted by C V E instead of the standard .E/ (to avoid confusion (Fig. A.1) with the already too numerous s appearing in Riemannian geometry: Christoffel symbols, harmonic n i n coordinates, ...).If U V is an open set, we let fx giD1 be local coordinates for @ @=@ i n 1. ; n/ the points p 2 U. The vector fields f i WD x giD1 2 C U TV provide the n local (positively oriented) coordinate basis for TpV , p 2 U. The corresponding n 1 dual basis for the cotangent spaces Tp V , is provided by the set of –forms i n 1. ; n/ 1. n; 2 n/ fdx giD1 2 C U T V . In particular, the metric tensor g 2 C V ˝S T V , 1. n; 2 n/ where C V ˝S T V denotes the set of smooth symmetric bilinear forms over n i k V , has the local coordinates representation g D gikdx ˝dx , where gik WD g.@i;@k/, and the Einsteinp summation convention is in effect. The RiemannianR volume form is 1 ::: n . n/ d g D det gdx ^ ^dx , and we denote by Volg V WD Vn d g the volume of (the compact) .Vn; g/. The distance between points x; y 2 Vn, characterizing .Vn; g/ as a metric space, is defined by setting © Springer International Publishing AG 2017 347 M. Carfora, A. Marzuoli, Quantum Triangulations, Lecture Notes in Physics 942, https://doi.org/10.1007/978-3-319-67937-2 348 A Riemannian Geometry Fig. A.1 ...Riemanniangeometrymayhaveunexpectedsides...andmustbeusedresponsibly! Z 1 p dg.x; y/ WD inf g..P t/; .P t// dt; (A.1) 2 C .x;y/ 0 where C .x; y/ is the space of (piecewise C1) curves W Œ0; 1 ! Vn such that .0/ D x and .1/ D y.If 2 k1:::k j1:::j jUj WD gi1h1 :::girhr U q U q g :::g (A.2) g i1:::ip hi:::hr k1j1 kqjq denotes the pointwise (squared) g–norm in the tensor bundle T.r;q/Vn WD ˝r TVn ˝q TVn, then we can endow the space of smooth .r; q/–tensor fields 1 n .r;q/ n n p p; s C .V ; T V / on V with the corresponding L .dg/ and H .dg/ Sobolev norms ÄZ 1 p : p p ;1 < ; jjUjjL .dg/ D jUjg d g Ä p 1 (A.3) Vn 1 s ÄZ X ˇ ˇ p : ˇ .i/ ˇ p H p; s. / ; jjUjj d g D r U g d g (A.4) n iD0 V where s is an integer 0, and r.i/U denotes the ith covariant derivative of the tensor field U. The completions of C1.Vn; T.r;q/Vn/ in these norms, define A Riemannian Geometry 349 Fig. A.2 The group of diffeomorphisms the Sobolev spaces of sections H p; s.Vn; T.r;q/Vn/. In particular, for p D 2,we denote by H s.Vn; T.r;q/Vn/ the Hilbert space H 2;s.Vn; T.r;q/Vn/. According to 0 > n H s. n; .r;q/ n/ Sobolev embedding theorem, if k and s 2 C k, then V T V Ck.Vn; T.r;q/Vn/. Similarly, if we denote by H s.Vn; Vn/ the Sobolev space of maps W Vn ! Vn, then H s.Vn; Vn/ Ck.Vn; Vn/ is a dense continuous inclusion for s >.n=2/ C k. Let ˚ ˇ « Diff k.Vn/ WD 2 Ck.Vn; Vn/ˇ is a bijection and 1 2 Ck.Vn; Vn/ (A.5) be the group of Ck diffeomorphisms of Vn, endowed with the compact–open Ck topology. The group of smooth diffeomorphisms Diff.Vn/ is defined by the sequence of natural inclusions (Fig. A.2) Diff 1.Vn/ Diff 2.Vn/ :::Diff k.Vn/ ::: ; (A.6) according to . n/ 1 k. n/: Diff V D\kD1 Diff V (A.7) Recall that under a diffeomorphism W Vn ! Vn the pull back from ..Vn/; g/ to .Vn;g/ is given for p 2 Vn by ˇ ˇ . ; / . ; / ; g V W p WD g V W j.p/ (A.8) 350 A Riemannian Geometry n where V denotes the push forward of V defined by the tangent map W TpV ! n i a i a i T.p/V . In local coordinates .U; x / and ..U/; y /, U 3 x 7! .x / WD ya..p// 2 .U/, we write @a @b g .p/ D g ..p//: (A.9) ik @xi @xk ab It is important to stress that the pull–back action of Diff.Vn/ on the metric g is a right action since under composition of any two diffeomorphisms and we have . ı /g D . g/. This latter remark plays a role when, for technical reasons, it is necessary to enlarge Diff.Vn/ to the (Hilbert manifold of the) diffeomorphisms of Sobolev class H s,fors >.n=2/ C 1, ˚ « D s.Vn/ WD 2 H s.Vn; Vn/j is a bijection and 1 2 H s.Vn; Vn/ : (A.10) In such a framework the group of smooth diffeomorphisms can be recovered from D s.Vn/ by characterizing the smooth topology on Diff.Vn/ as the limit of the topologies of D s.Vn/, by viewing Diff.Vn/ as an Inverse Limit Hilbert group in n s n the sense of Omori [26], i.e. Diff.V / D\s>n=2D .V /. Since right multiplication is smooth, but left multiplication is not, D s.Vn/ is not “an infinite–dimensional” Lie group but only a topological group [11,21]. A.2 Curvature and Scaling Besides diffeomorphisms, the metric g is naturally acted upon also by overall rescalings according to for 2 R>0 (Fig. A.3) g 7! g (A.11) Fig. A.3 Besides diffeomorphisms, the metric g is naturally acted upon also by overall rescalings A Riemannian Geometry 351 g1 7! 1 g1; 1 1. n; 2 n/ where g 2 C V ˝S TV denotes the inverse metric providing the inner n n . ; i/ product in Tp V , p 2 V . In local coordinates U x , we write gik 7! gik and gik 7! 1 gik, respectively. Under the scaling (A.11) we immediately get an obvious dilation or contraction effect on distances and volume 1 2 dg.p; q/ 7! dg.p; q/ D dg.p; q/ n (A.12) n n 2 n Vol g.V / 7! Vol g.V / D Vol g.V /; which is often useful to factor out in the study of Riemannian functional. In this connection it is worthwhile to recall how the basic Riemannian definitions interact with (A.11). We start with the Levi–Civita connection rg of .Vn; g/,(“r” if there is no danger of confusion), defined by 2 g .rX Y; Z/ WD X .g.Y; Z// C Y .g.Z; X// Z .g.X; Y// (A.13) C g .ŒX; Y; Z/ g .ŒY; Z; X/ C g .ŒZ; X; Y/ ; along with the torsion–free and metric compatibility conditions rXY rY X D ŒX; Y (A.14) X .g.Y; Z// D g .rXY; Z/ C g .Y; rXZ/ ; for all vector fields X; Y; Z 2 C1.Vn; TVn/. In a local coordinate neighborhood . ; i / @ @ k. /@ k. / U fx g we write ri j WD r@i j WD ij g k, where the Christoffel symbols ij g are given by  à 1 @g ` @g` @g k.g/ D gk` j C i ij : (A.15) ij 2 @xi @xj @x` If .Vn;g/ is the pull–back of ..Vn/; g/ under a diffeomorphism W Vn ! Vn, k. / k. / then the Christoffel symbols ij g and ij g are related by @c @a @b @2c k.g/ D c .g/ ı C ; (A.16) ij @xk ab @xi @xj @xi@xj whereas, under the metric rescaling (A.11), we easily compute from (A.15) k. / k. / ij g D ij g , hence r g Drg: (A.17) 1 n Let us recall that if Hessg f WD rdf denotes the Hessian of f , f 2 C .V ; R/, then n the Laplace–Beltrami operator 4g on .V ; g/ is defined by 352 A Riemannian Geometry ik 4g f WD trg Hessg f D g rirk f ; (A.18) where we have used the geometer’s sign convention according to which, on P 2 .Rn;ı/ n @ Euclidean space , 4ı D iD1 @ 2 . It is useful to have the expression of xi 4g available in local coordinates  à 1 @ p @ ij 4g D p det gg det g @xi @xj @2 1 @ p Á @ (A.19) D gij C p det ggij @xi@xj det g @xi @xj @2 @ D gij j ; @xi@xj @xj where we have defined @ @ p Á j WD ghk j D gji gij log det g : (A.20) hk @xi @xi From the definitions of Hessg and 4g it immediately follows that under the metric rescaling (A.11), we have Hessg D Hessg; (A.21) 1 4 g D 4g: The Riemann, the sectional, the Ricci and the scalar curvature operators (Fig.
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