Lecture Notes in MathematicsEditors:J.-M.Morel,CachanF.Takens,GroningenB.Teissier,Paris1798
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Waldyr Muniz OlivaGeometric Mechanics1 3
AuthorWaldyr Muniz OlivaCAMGSD,Departamento de MatemáticaInstituto Superior TécnicoAv.Rovisco Pais1049-001 Lisboa,PortugalE-mail: wamoliva@math.ist.utl.pt, wamoliva@ime.usp.brCataloging-in-Publication Data applied forMathematics Subject Classification (2000):70Hxx, 70G45, 37J60ISSN 0075-8434ISBN 3-540-44242-1 Springer-Verlag Berlin Heidelberg NewYorkThis work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965,in its current version,and permission for use must always be obtained from Springer-Verlag.Violations areliable for prosecution under the German Copyright Law.Springer-Verlag Berlin Heidelberg NewYork a member of BertelsmannSpringerScience + Business Media GmbHhttp://www.springer.de© Springer-Verlag Berlin Heidelberg 2002Printed in GermanyThe use of general descriptive names,registered names,trademarks,etc.in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective lawsand regulations and therefore free for general use.Typesetting: Camera-ready TEX output by the authorSPIN: 1089103941/3142/du-543210 - Printed on acid-free paperBibliographic information published by Die Deutsche BibliothekDie Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic data is available in the Internet at http://dnb.ddb.de
To my wife Angela for inspiration and constant support
PrefaceThis book was based on notes which were prepared as a guide for lecturesof one semester course on Geometric Mechanics. They were written insidethe level of a master course. I started some years ago teaching them at the“Instituto de Matem´ atica e Estat´istica” of the “Universidade de S˜ ao Paulo”,and, more recently, at the “Instituto Superior T´ ecnico” of the “UniversidadeT´ ecnica de Lisboa”.The spectrum of participants of such a course ranges usually from youngMaster students to Phd students. So, it is always very difficult to decide howto organize all material to be taught. I decided that the expositions shouldbe self contained, so some subjects that one expects to be interesting forsomeone, result, often, tedious for others and frequently unreachable for afew ones.In any case, for young researchers interested in differential geometry andor dynamical systems, it is basic and fundamental to see the foundationsand the development of classical subjects like Newtonian and RelativisticMechanics.I wish to thank a number of colleagues from several different Institutionsas well as Master and PhD students from S˜ ao Paulo and Lisbon who moti-vated and helped me with comments and suggestions when I was writing thistext. Among them I mention Jack Hale, Ivan Kupka, Giorgio Fusco, PauloCordaro, Carlos Rocha, Luis Magalh˜ aes, Luis Barreira, Esmeralda Dias, Za-queu Coelho, Helena Castro, Marcelo Kobayashi, S´ onia Garcia, Diogo Gomesand Jos´ e Nat´ ario. I am also very grateful to Ms. Achi Dosanjh of Springer-Verlag for her help and encouragement; it has been a pleasure working withher and her Springer-Verlag colleagues. Thanks are also due to Ana Bor-dalo for her fine typing of this work and to FCT (Portugal) for the supportthrough the program POCTI.Lisbon, May 2002Waldyr Muniz Oliva
Table of ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1Embedded manifolds in RN1.2The tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3The derivative of a differentiable function . . . . . . . . . . . . . . . . .1.4Tangent and cotangent bundles of a manifold . . . . . . . . . . . . . .1.5Discontinuous action of a group on a manifold . . . . . . . . . . . . .1.6Immersions and embeddings. Submanifolds . . . . . . . . . . . . . . . .1.7Partition of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .347899. . . . . . . . . . . . . . . . . . . . . . . . . . . . .11122Vector fields, differential forms and tensor fields . . . . . . . . . .2.1Lie derivative of tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2The Henri Cartan formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1316203Pseudo-Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1Affine connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2The Levi-Civita connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3Tubular neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5E. Cartan structural equations of a connection . . . . . . . . . . . . .25293436504Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1Galilean space-time structure and Newton equations . . . . . . . .4.2Critical remarks on Newtonian mechanics . . . . . . . . . . . . . . . . .55605Mechanical systems on Riemannian manifolds . . . . . . . . . . . . 615.1The generalized Newton law . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2The Jacobi Riemannian metric . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3Mechanical systems as second order vector fields. . . . . . . . . . . .5.4Mechanical systems with holonomic constraints . . . . . . . . . . . .5.5Some classical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.6The dynamics of rigid bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.7Dynamics of pseudo-rigid bodies. . . . . . . . . . . . . . . . . . . . . . . . . . 1025.8Dissipative mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 107616366687078
XTable of Contents6Mechanical systems with non-holonomic constraints . . . . . . 1116.1D’Alembert principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2Orientability of a distribution and conservation of volume . . . 1196.3Semi-holonomic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4The attractor of a dissipative system . . . . . . . . . . . . . . . . . . . . . . 1237Hyperbolicity and Anosov systems. Vakonomic mechanics 1277.1Hyperbolic and partially hyperbolic structures . . . . . . . . . . . . . 1277.2Vakonomic mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.2.1Some Hilbert manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.2.2Lagrangian functionals and D-spaces. . . . . . . . . . . . . . . . 1367.3D’Alembert versus vakonomics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.4Study of the D–spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.4.1The tangent spaces of H1(M,D,[a0,a1],m0) . . . . . . . . . 1377.4.2The D–space H1(M,D,[a0,a1],m0,m1). Singular curves1407.5Equations of motion in vakonomic mechanics. . . . . . . . . . . . . . . 1428Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.1Lorentz manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2The quadratic map of Rn+11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.3Time-cones and time-orientability of a Lorentz manifold . . . . . 1508.4Lorentz geometry notions in special relativity . . . . . . . . . . . . . . 1538.5Minkowski space-time geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.6Lorentz and Poincar´ e groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1629General relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659.1Einstein equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659.2Geometric aspects of the Einstein equation . . . . . . . . . . . . . . . . 1669.3Schwarzschild space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1699.4Schwarzschild horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.5Light rays, Fermat principle and the deflection of light . . . . . . 175AHamiltonian and Lagrangian formalisms . . . . . . . . . . . . . . . . . . 183A.1 Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183A.2 Euler–Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185BM¨ obius transformations and the Lorentz group . . . . . . . . . . . 195B.1 The Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195B.2 Stereographic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199B.3 Complex structure of S2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201B.4 M¨ obius transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204B.5 M¨ obius transformations and the proper Lorentz group . . . . . . 207B.6 Lie algebra of the Lorentz group. . . . . . . . . . . . . . . . . . . . . . . . . . 211B.7 Spinors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215B.8 The sky of a rapidly moving observer . . . . . . . . . . . . . . . . . . . . . 217
Table of ContentsXICQuasi-Maxwell form of Einstein’s equation. . . . . . . . . . . . . . . . 223C.1 Stationary regions, space manifold and global time. . . . . . . . . . 223C.2 Connection forms and equations of motion . . . . . . . . . . . . . . . . . 226C.3 Stationary Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230C.4 Curvature forms and Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . 231C.5 Quasi-Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235C.6 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239DViscosity solutions and Aubry–Mather theory . . . . . . . . . . . . 245D.1 Optimal control and time independent problems . . . . . . . . . . . . 245D.2 Hamiltonian systems and the Hamilton–Jacobi theory . . . . . . 249D.3 Aubry–Mather theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
IntroductionGeometric Mechanics in this book means Mechanics on a pseudo-rieman-nian manifold and the main goal is the study of some mechanical modelsand concepts, with emphasis on the intrinsic and geometric aspects arisingin classical problems. Topics like calculus of variation and the theories ofsymplectic, Hamiltonian and Poissonian structures including reduction bysymmetries, integrability etc., also related with most of the considered mod-els, were avoided in the body because they already appear in many modernbooks on the subject and are also contained in other courses of the majorityof Master and PhD programs of many Institutions (see [1], [27], [46], [47]).The first seven chapters are written under the spirit of Newtonian Me-chanics while the two last ones describe the foundations and some aspectsof Special and General Relativity. They have a coordinate free presentationbut, for a sake of motivation, many examples and exercises are included inorder to exhibit the desirable flavor of physical applications. In particular,some of them show, for instance, numerical differences appearing between theNewtonian and relativistic formulations.Chapters 1 and 2 include the fundamental calculus on a differentiablemanifold with a brief introduction of vector fields, differential forms and ten-sor fields. Chapter 3 starts with the concept of affine connection and specialattention is given to the notion of curvature; E. Cartan structural equationsof a connection are also derived in Chapter 3. Chapter 4 starts with the for-mulation of classical Newtonian mechanics where it is described the Galileanspace-time structure and Newton equations. Chapter 5 deals with mechanicalsystems on a Riemannian manifold including classical examples like the dy-namics of rigid and pseudo-rigid bodies; notions derived from dissipation inmechanics and, correspondingly, structural stability with generic properties ofthese (Morse–Smale) systems are also discussed. Chapter 6 considers mechan-ical systems with non-holonomic constraints and describes D’Alembertian ge-ometric mechanics including conservative and dissipative situations. In Chap-ter 7 one talks about hyperbolicity and Anosov systems arising in mechanicsand it is also mentioned the so-called non-holonomic mechanics of vakonomictype.In the end of Chapter 4 we present some critical remarks on the bases ofNewtonian Mechanics in order to motivate the introduction of Chapters 8 andW.M. Oliva: LNM 1798, pp. 1–2, 2002.c ? Springer-Verlag Berlin Heidelberg 2002
2Introduction9 on Special and General Relativity, respectively. To clarify and give sense tosome expressions and concepts usually found in Hamiltonian and Lagrangiantheories, freely used in previous chapters, it is introduced Appendix A with ashort presentation on Hamilton and Lagrange systems as well as few resultson the variational approach of classical mechanics. The book follows withAppendices B and C, written by Jos´ e Nat´ ario, where are discussed Lorentzgroup and the quasi-Maxwell form of Einstein’s equation, appearing as acomplement to Chapters 8 and 9. Finally Appendix D, written by DiogoGomes, deals with viscosity solutions and Aubry–Mather theory showing alsothe flavor of new areas related to Geometric Mechanics.
1 Differentiable manifoldsA topological manifoldspace with a countable basis of open sets such that each x ∈ Q has an openneighborhood homeomorphic to an open subset of the Euclidean space Rn.Each pair (U,ϕ) where U is open in Rnand ϕ is a homeomorphism of Uonto the open set ϕ(U) of Q is called a local chart, ϕ(U) is a coordinateneighborhood and the inverse ϕ−1: ϕ(U) −→ U, given by y ∈ ϕ(U) ?→ϕ−1(y) = (x1(y),...,xn(y)), is called a local system of coordinates. If apoint x ∈ Q is associated to two local charts ϕ : U −→ Q and ϕ : U −→ Q,that is x ∈ ϕ(U)∩ϕ(U), one obtains the bijection ϕ−1◦ϕ : W −→ W wherethe open sets W ⊂ U and W ⊂ U are given byW = ϕ−1?ϕ(U) ∩ ϕ(U)?Q of dimension n is a topological HausdorffandW = ϕ−1?ϕ(U) ∩ ϕ(U)?(U)ϕU( )????????????????????????????????????????????????????????????????????????U????????????????????????????????????????????????????????????????????????????????URnRn????????????????????????????????????????????????????????????????????????????????ϕϕϕQFig. 1.1. Two intersecting charts on a topological manifold.W.M. Oliva: LNM 1798, pp. 3–12, 2002.c ? Springer-Verlag Berlin Heidelberg 2002
41 Differentiable manifoldsThe charts (ϕ,U) and (ϕ,U) are said to be Ck- compatible if ϕ−1◦ ϕ :W −→ W is a Ck-diffeomorphism, k ≥ 1, k = ∞ or k = ω.A Ck-atlas is a set of Ckcompatible charts covering Q. Two Ck-atlasesare said to be equivalent if their union is a Ck-atlas. A Ck(differentiablemanifold) is a topological manifold Q with a class of equivalence of Ck-atlases. A manifold is connected if it cannot be divided into two disjointopen subsets (if no mention is made, a manifold means a C∞-differentiablemanifold).Examples of differentiable manifolds:Example 1.0.1. RnExample 1.0.2. The sphere S2= {(x,y,z) ∈ R3|x2+ y2+ z2= 1}.Example 1.0.3. The configuration space S1of the planar pendulum.Example 1.0.4. The configuration space of the double planar pendulum, thatis, the torus T2= S1× S1.Example 1.0.5. The configuration space of the double spherical pendulum,that is, the product S2× S2of two spheres.Example 1.0.6. The configuration space of a “rigid” line segment in the plane,R2× S1.Example 1.0.7. The configuration space of a “rigid” right triangle AOB,ˆO =90◦, that moves around O; it can be identified with the set SO(3) of all 3×3orthogonal matrices with determinant 1.Example 1.0.8. Pn(R), the n-dimensional real projective space (set of linespassing through 0 ∈ Rn+1), n ≥ 1.1.1 Embedded manifolds in RNWe say that Qn⊂ RNis a Cksubmanifold of (manifold embedded in)RNwith dimension n ≤ N, if Qnis covered by a finite or countable number ofimages ϕ(U) of the so called regular parametrizations , that is, Ck-maps,k ≥ 1,ϕ : U ⊂ Rn−→ RN, U open set of Rn, such that:i) ϕ : U −→ ϕ(U) is a homeomorphism where ϕ(U) is open in Qnwith thetopology induced by RN;ii)∂ϕ∂x(x0) : Rn−→ RNis injective for all x0∈ U.
1.1 Embedded manifolds in RN5x0RN(U)????????????????ϕx0( )ϕ????????????????????????????????????????????????????????????????????????????????????????Qn????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????ϕURnFig. 1.2. Manifold embedded in RN.Here ϕ(x1,...,xn) = (ϕ1(x1,...,xn),...,ϕN(x1,...,xn)) andN × n matrixTo show that Qnis a Ckmanifold we prove the next two propositions:∂ϕ∂x(x0) is the∂ϕ∂x(x0) = (∂ϕi∂xj(x0)).Proposition 1.1.1. Let Qnbe a Cksubmanifold of RNwith dimension nand ϕ : U −→ RNa regular parametrization in a neighborhood of y0 ∈ϕ(U) ⊂ Qn. Then, there exist an open neighborhood Ω of y0 in RNand aCk-map Ψ : Ω −→ RNsuch thatΨ (Qn∩ Ω) = ϕ−1(Qn∩ Ω) × {0}N−n.Proof:We may assume, without loss of generality, that the first deter-minant (n first lines and n columns) ofy0 = ϕ(x0)). Define the function F : U × RN−n−→ RNby F(x;z) =(ϕ1(x),...,ϕn(x);ϕn+1(x) + z1,...,ϕN(x) + zN−n) which is of class Ck; wehave, clearly, F(x,0) = ϕ(x), for all x ∈ U, so F(x0,0) = ϕ(x0) = y0andi,j = 1,...,n...∂ϕ∂x(x0) does not vanish (here∂F∂(x,z)(x0,0) =∂ϕi∂xj(x0)...0................ ......∗I.From this it follows that detinverse function theorem, that is, F is a (local) diffeomorphism onto an open∂F∂(x,z)(x0,0) ?= 0. The result comes, using the
61 Differentiable manifolds????????????????????????????????????????????????????????????????y =0( )ϕ x0x0(U)ϕπ1π2N−nQn??????????ϕURRΩΨFnNU X RFig. 1.3. Proof of Proposition 1.1.1.neighborhood Ω of y0in RNwith an inverse Ψ defined in Ω. It is also clearthat Ψ (Qn∩ Ω) = ϕ−1(Qn∩ Ω) × {0}N−n.From the last proposition it follows that any Cksubmanifold is in fact aCkmanifold.Remark 1.1.2. Denote by π2the second projection π2: U ×RN−n−→ RN−nand let f be the composition f = π2◦ Ψ : Ω −→ RN−n, so that, to anyy ∈ Qnthat belongs to Ω, one associates N − n functions f1,...,fN−n :Ω −→ R such that f = (f1,...,fN−n) and Ω ∩Qnis given by the equationsf1 = ... = fN−n = 0, the differentials df1(y),...,dfN−n(y) being linearlyindependent.Conversely we have the following:Proposition 1.1.3. Let Q ⊂ RNbe a set such that any point y ∈ Q has anopen neighborhood Ω in RNand N − n Ck-differentiable functions, k ≥ 1,f1: Ω −→ R,...,fN−n: Ω −→ Rsuch that Ω ∩ Q is given by f1= ... = fN−n= 0, with df1(y),...,dfN−n(y)linearly independent. Then Q is a Cksubmanifold of (manifold embedded in)RNwith dimension n.Proof:surjective linear transformationThe linear forms dfi(y) : RN−→ R, i = 1,...,N − n, define a
1.2 The tangent space7(df1(y),...,dfN−n(y)) : RN−→ RN−nwith a n-dimensional kernel K ⊂ RN. Let L : RN−→ Rnbe any lineartransformation such that the restriction L|K is an isomorphism from K ontoRn. Define G : Ω ⊂ RN−→ RNbyG(ξ) = (f1(ξ),...,fN−n(ξ),L(ξ))whose derivative at y ∈ Q is given bydG(y)v = (df1(y)v,...,dfN−n(y)v,L(v)).Then dG(y)v is non singular and so, by the inverse function theorem, Gtakes an open neighborhood˜Ω of y, diffeomorphically onto a neighborhoodG(˜Ω) of (0,L(y)). Note that if f= (f1,...,fN−n), f−1(0) ∩˜Ω = Q ∩˜Ωcorresponds, under the action of G, to points of the hyperplane (0,Rn) sinceG takes f−1(0) ∩˜Ω onto (0,Rn) ∩ G(˜Ω). The inverse ϕ of G restricted tof−1(0) ∩˜Ω is a Ck-bijection:= (0,Rn) ∩ G(˜Ω) −→ ϕ(U) = Q ∩To the point y ∈ Q then corresponds a local chart (ϕ,U), that is, Q is aCk-submanifold of RN, with dimension n.defϕ : Udef˜Ω.Exercise 1.1.4. The orthogonal matrices are obtained between the real 3×3matrices (these are essentially R9) as the zeros of six functions (the orthogo-nality conditions). This way we obtain two connected components, since thedeterminant of an orthogonal matrix is equal to +1 or −1. The componentwith determinant +1 is the group SO(3) of rotations of R3. Show that SO(3)is a compact submanifold of R9of dimension 3.1.2 The tangent spaceLet Q be a n-dimensional submanifold of RN. To any y ∈ Q is associated asubspace TyQ of dimension n; in the notation of Proposition 1.1.3, TyQ isthe kernel K of the linear map(df1(y),...,dfN−n(y)) : RN−→ RN−n.The vectors of TyQ = K are called the tangent vectors to Q at thepoint y ∈ Q and the subspace TyQ is the tangent space of Q at the pointy. The tangent vectors at y can also be defined as the velocitiesC1-curves γ : (−ε,+ε) −→ RNwith values on Q and such that γ(0) = y.In the general case of a manifold Q one defines an equivalence relation aty ∈ Q between smooth curves. So, a continuous curve γ : I −→ Q (I is any·γ(0) of all
81 Differentiable manifoldsinterval containing 0 ∈ R) is said to be smooth at zero if for any local chart(U;ϕ),γ(0) = y ∈ ϕ(U), the curve ϕ−1◦ γ|γ−1(ϕ(U)): γ−1(ϕ(U)) −→ Rnissmooth. Two (smooth at zero) curves γ1: I1−→ Q and γ2: I2−→ Q suchthat γ1(0) = γ2(0) = y are equivalent ifThis concept does not depend on the local chart (U;ϕ). A tangent vectorvyat y ∈ Q is a class of equivalence of that equivalence relation. We writesimply vy =product of a real number by a tangent vector. This way the set TyQ of alltangent vectors to Q at y ∈ Q is a vector space with dimension n. Witha local chart (U;ϕ) and the canonical basis {ei}(i = 1,...,n) of Rn, it ispossible to construct a basis of TyQ at y ∈ ϕ(U); if we set x0 = ϕ−1(y),consider the tangent vectors associated to the curves γi: t ?→ ϕ(x0+tei) andlet=ddt(ϕ−1◦ γ1)|t=0=ddt(ϕ−1◦ γ2)|t=0.·γ1 (0) =·γ2 (0). One defines sum of tangent vectors at y and∂∂xi(y)def·γi(0),i = 1,...,n.1.3 The derivative of a differentiable functionA continuous function f : Q1−→ Q2defined on a differentiable manifold Q1with values on a differentiable manifold Q2is said to be Cr- differentiable aty ∈ Q1if for any two charts (U,ϕ) and (U,ϕ), ϕ−1(y) ∈ U and ϕ−1(f(y)) ∈U, the map ϕ−1.f.ϕ : U −→ U is Crdifferentiable at ϕ(y), r > 0; of coursewe are assuming (as we can) f(ϕ(U)) ⊂ ϕ(U) (reducing U if necessary), dueto the continuity of f at y ∈ Q1. The notion of differentiability does notdepend on the used local charts. One uses to say smooth instead of C∞.The derivative df(y) or f∗(y) of a C1- differentiable function f : Q1−→ Q2at y ∈ Q1is a linear mapf∗(y) : TyQ1−→ Tf(y)Q2that sends a tangent vector represented by a curve γ : I −→ Q1, γ(0) =y ∈ Q1, into the tangent vector at f(y) ∈ Q2 represented by the curvefoγ : I −→ Q2. One can show that f∗(y) is linear.If g : Q2−→ Q3is another C1-differentiable function one has:f∗(y)−→ Tf(y)Q2and it can be proved thatTyQ1g∗(f(y))−→Tg(f(y))Q3(g ◦ f)∗(y) = g∗(f(y)) ◦ f∗(y)A Cr-diffeomorphism f : Q1−→ Q2is a bijection such that f and f−1are Cr-differentiable, r ≥ 1 .for ally ∈ Q1.
1.5 Discontinuous action of a group on a manifold91.4 Tangent and cotangent bundles of a manifoldLet Q be a Ck-differentiable manifold, k ≥ 2. Consider the sets TQ =∪y∈Q TyQ and T∗Q = ∪y∈QTy∗Q where Ty∗Q, y ∈ Q, is the dual of TyQ,that is, Ty∗Q is the set of all linear forms defined on TyQ.Exercise 1.4.1. Show that TQ and T∗Q are Ck−1-manifolds if Q is a Ck-manifold, k ≥ 2. Show also that the canonical projections:τ : vy∈ TQ ?→ y ∈ Qτ∗: ωy∈ T∗Q ?→ y ∈ Qare Ck−1maps.TQ and T∗Q are called the tangent and cotangent bundles of Q,respectively.andExercise 1.4.2. Prove that the cartesian product of two manifolds is a man-ifold.Exercise 1.4.3. (Inverse image of a regular value) Let F : U ⊂ Rn−→ Rmbe a differentiable map defined on an open set U ⊂ Rn. A point p ∈ U isa critical point of F if dF(p) : Rn−→ Rmis not surjective. The imageF(p) ∈ Rmof a critical point is said to be a critical value of F. A pointa ∈ Rmis a regular value of F if it is not a critical value. Show that theinverse image F−1(a) of a regular value a ∈ Rmeither is a submanifold ofRn, contained in U, with dimension equal to n − m, or F−1(a) = ∅.Let Q be a differentiable manifold. Q is said to be orientable if Qhas an atlas a = {(Uα,ϕα)} such that (Uα,ϕα) and (Uβ,ϕβ) in a sat-isfying ϕα(Uα) ∩ ϕβ(Uβ) ?= ∅, the derivative of ϕβ−1◦ ϕα at any x ∈ϕα−1[ϕα(Uα) ∩ ϕβ(Uβ)] has positive determinant. If one fix such an atlas, Qis said to be oriented. If Q is orientable and connected, it can be orientedin exactly two ways.Exercise 1.4.4. Show that TQ is orientable (even if Q is not orientable).Show that a two-dimensional submanifold Q of R3is orientable if, and onlyif, there is on Q a differentiable normal unitary vector field N : Q −→ R3,that is, for all y ∈ Q,N(y) is orthogonal to TyQ.Exercise 1.4.5. Use the stereographic projections and show that the sphereSn= {(x1,...,xn+1) ∈ Rn+1|?n+11.5 Discontinuous action of a group on a manifoldi=1x2i= 1} is orientable.An action of a group G on a differentiable manifold M is a mapϕ : G × M −→ Msuch that:
101 Differentiable manifolds1) for any fixed g ∈ G, the map ϕg: M −→ M given by ϕg(p) = ϕ(g,p) isa diffeomorphism and ϕe= Identity on M (e ∈ G is the identity);2) if g and h are in G then ϕgh= ϕg◦ ϕhwhere gh is the product in G.An action ϕ : G × M −→ M is said to be properly discontinuous ifany p ∈ M has a neighborhood Upin M such that Up∩ ϕg(Up) = ∅ for allg ?= e, g ∈ G.Any action of G on M defines an equivalence relation ∼ between elementsof M; in fact, one says that p1∼ p2(p1equivalent to p2) if there exists g ∈ Gsuch that ϕg(p1) = p2. The quotient space M/G under ∼ with the quotienttopology is such that the canonical projection π : M −→ M/G is continuousand open. (π(p) ∈ M/G is the class of equivalence of p ∈ M).The open sets in M/G are the images by π of open sets in M. Since Mhas a countable basis of open sets, M/G also has a countable basis of opensets.Exercise 1.5.1. Show that the topology of M/G is Hausdorff if and only ifgiven two non equivalent points p1,p2 in M, there exist neighborhoods U1and U2of p1and p2such that U1∩ ϕg(U2) = φ for all g ∈ G.Exercise 1.5.2. Show that if ϕ : G×M −→ M is properly discontinuous andM/G is Hausdorff then M/G is a differentiable manifold and π : M −→ M/Gis a local diffeomorphism, that is, any point of M has an open neighborhoodΩ such that π sends Ω diffeomorphically onto the open set π(Ω) of M/G.Show also that M/G is orientable if and only if M is oriented by an atlas a ={(Uα,ϕα)} preserved by the diffeomorphisms ϕg,g ∈ G (that is, (Uα,ϕg◦ϕα)belongs to a for all (Uα,ϕα) ∈a).Example 1.5.3. Let M = Sn⊂ Rn+1and G be the group of diffeomorphismsof Snwith two elements: the identity and the antipodal map A : x ?→ −x.The quotient Sn/G can be identified with the projective space Pn(R).Example 1.5.4. Let M = Rkand G be the group Zkof all integer transla-tions, that is, the action of g = (n1,...,nk) ∈ Zkon x = (x1,...,xk) ∈ Rkmeans to obtain x + g ∈ Rk. The quotient Rk/Zkis the torus Tk. Thetorus T2is diffeomorphic to the torus of revolution˜T2, submanifoldof R3obtained as the inverse image of zero under the map f(x,y,z) =z2+ (?x2+ y2− a)2− r2(0 < r < a).Example 1.5.5. Let S be a submanifold of R3symmetric with respect tothe origin and G = {e,A} be the group considered in example 1.5.3 above.The special case S =˜T2(torus of revolution in R3) gives us the quotientmanifold˜T2/G= K, the so called Klein bottle. When S is the manifoldS = {(x,y,z) ∈ R3|x2+y2= 1,−1 < z < 1} then S/G is called the M¨ obiusband.defExercise 1.5.6. Show that the Klein bottle, the M¨ obius band and P2(R)are not orientable. Show also that Pn(R) is orientable if and only if n is odd.
1.6 Immersions and embeddings. Submanifolds111.6 Immersions and embeddings. SubmanifoldsLet M and N be differentiable manifolds and ϕ : M −→ N be a differentiablemap. ϕ is said to be an immersion of M into N if ϕ∗(p) : TpM −→ Tϕ(p)Nis injective for all p ∈ M.An embedding of M into N is an immersion ϕ : M −→ N such that ϕis a homeomorphism of M onto ϕ(M) ⊂ N, ϕ(M) with the topology inducedby N. If M ⊂ N and the inclusion i : M −→ N is an embedding of M intoN, M is said to be a submanifold of N.Example 1.6.1. The map ϕ : R −→ R2given by ϕ(t) = (t,|t|) is not differen-tiable at t = 0.Example 1.6.2. The map ϕ : R −→ R2defined by ϕ(t) = (t3,t2) is differen-tiable but is not an immersion because ϕ∗(0) : R −→ R2is the zero map thatis not injective.Example 1.6.3. The map ϕ : (0,2π) −→ R2defined byϕ(t) = (2cos(t −πis an immersion but is not a embedding. The image M = ϕ((0,2π)) is an”eight”. Also, the inclusion i : M −→ R2is not an embedding, so M =ϕ((0,2π)) is not a submanifold of R2.Example 1.6.4. The curve ϕ : (−3,0) −→ R2given by:is an immersion but is not an embedding.A neighborhood of O = (0,0) has infinitely many connected componentsif one considers the induced topology for the set ϕ(−3,0) ⊂ R2.Example 1.6.5. ϕ : R −→ R3defined by ϕ(t) = (cos2πt,sin2πt,t) is anembedding. The image ϕ(R) is homeomorphic to R.Example 1.6.6. The image ϕ(R) of the map ϕ : R −→ R2given by ϕ(t) =(cos2πt,sin2πt) is S1⊂ R2. The map ϕ is an immersion but not an em-bedding since is not injective. But ϕ(R) = S1is a submanifold of R2if weconsider the inclusion map i : S1−→ R2.Exercise 1.6.7. Analyze the maps:2),sin2(t −π2))ϕ(t) =(0,−(t + 2)) ifa regular curve for t ∈ (−1,−1(−t,−sin1t ∈ (−3,−1)t ∈ (−1π)t) ifπ,0)ϕ1(t) = (1ϕ2(t) = (t + 1tcos2πt,1tsin2πt),t ∈ (1,∞);sin2πt),2tcos2πt,t + 12πt ∈ (1,∞)
121 Differentiable manifolds1.7 Partition of unityLet X be a topological space. A covering of X is a family {Ui} of open sets Uiin X such that?the covering, only. One says that a covering {Vk} is subordinated to {Ui} ifeach Vkis contained in some Ui. Let Brbe the ball of Rmcentered at 0 ∈ Rmand radius r > 0.iUi= X. A covering of X is said to be locally finite if anypoint of X has a neighborhood that intersects a finite number of elements inProposition 1.7.1. Let X be a differentiable manifold, dim X = m. Givena covering of X, there exists an atlas {(ϕ−1finite covering of X subordinated to the given covering, and such that ϕ−1is the ball B3and, moreover, the open sets Wk= ϕk(B1) cover X.k(Vk),ϕk)} where {Vk} is a locallyk(Vk)For a proof see the book [40] ”Differential Manifolds” by S. Lang, AddisonWesley Pu. Co., p. 33, taking into account that in the last Proposition 1.7.1X is Hausdorff, finite dimensional and has a countable basis.The support supp (f) of a function f : X → R is the closure of the setof points where f does not vanish. We say that a family {fk} of differentiablefunctions fk: X → R is a differentiable partition of unity subordinatedto a covering {Vk} of X if:(1) For any k, fk≥ 0 and supp (fk) is contained in a coordinate neighbor-hood Vkof an atlas {(ϕ−1(2) The family {Vk} is a locally finite covering of X.(3)p, fα(p) ?= 0 for a finite number of indices, only).Proposition 1.7.2. Any connected differentiable manifold X has a differ-entiable partition of unity.k(Vk),ϕk)} of X.?αfα(p) = 1 for any p ∈ X (this condition makes sense since for eachProof: The idea is the following: by Proposition 1.7.1, for each k one definesa smooth “cut off” function ψk: X → R of compact support contained inVk such that ψk is identically 1 on Wk and ψk ≥ 0 on X. From the factthat {Vk} is a locally finite coveri...