DOCSLIB.ORG

Differential Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Differential Topology Differential Topology: Homework Set # 3. 1) Basic on tangent bundles. Recall that the tangent bundle TM to an extrinsicly defined manifold M k ⊂ Rn is defined to be: n n k n ∼ n TM := {(p, v) ∈ R × R | p ∈ M , v ∈ TpM ⊂ TpR = R } This exercise is designed to give you some practice working with tangent bundles. (i) Show that for a manifold M, the tangent bundle TM is well-defined, i.e. does not depend on the embedding M k ⊂ Rn. (ii) If M, N are a pair of smooth manifolds, show that T (M × N) is diffeomorphic to TM × TN. (iii) If f : M → R is a smooth, positive function, show that the corresponding scaling map φ : TM → TM defined by φ(p, v) = (p, f(p) · v) is a diffeomorphism. (iv) Show that the tangent bundle to S1 (the unit sphere) is diffeomorphic to the cylinder S1 × R. (v) Show that if Sn is an arbitrary sphere, then there exists a diffeomorphism between (TSn) × R and Sn × Rn+1. 2) Relatives of the tangent bundle. Closely associated to the tangent bundle are some other manifolds: (i) For an extrinsicly defined manifold M k ⊂ Rn, define the normal bundle as follows: n n k ⊥ n ∼ n N(M) := {(p, v) ∈ R × R | p ∈ M , v ∈ TpM ⊂ TpR = R } Show that N(M) is an n-dimensional manifold. (ii) Show that for the standard n-sphere Sn ⊂ Rn+1, the normal bundle N(Sn) is diffeomorphic to Sn × R. (iii) Consider the subset T 1M of TM consisting of pairs (p, v) ∈ TM with ||v|| = 1 (where || · || denotes the standard norm on Rn). Show that T 1M is a (2k − 1)-dimensional manifold, called the sphere bundle of M (or the unit tangent bundle of M). [Hint: construct a suitable submersion from TM → R.] (iv) Recall that a vector field on M was defined to be a smooth map v : M → TM satisfying π ◦ v = IdM . Let ρ : Rn × Rn → Rn be the projection onto the second factor, and say that a vector field v has a zero at p if (ρ ◦ v)(p) = 0 ∈ Rn. Show that if Sn is an odd dimensional sphere, then there exists a non-vanishing vector field v on Sn (i.e. v has no zeros). (v) Show that if Sn supports a non-vanishing vector field v, then its antipodal map (i.e. p 7→ −p) is smoothly homotopic to the identity map. [Hint: normalize the v so that the image lies in T 1Sn, and rotate p to −p along the direction given by v(p).] 3) Intrinsic Whitney embedding. In class, I outlined an argument for showing the intrinsic version of Whit- ney embedding, using some notions from differential geometry. This exercise outlines an alternate, “bare hands” construction of an embedding of the compact manifold M k into some RN (with large N). (i) Show that the function f : R → R defined below is smooth: ( 2 e−1/x x > 0 f(x) = 0 x ≤ 0. (ii) Given a < b, show that the associated function g(x) = f(x − a)f(b − x) is smooth, positive on the interval (a, b) ⊂ R and identically zero elsewhere. Show that the corresponding function h : R → R defined by: R t −∞ g dx h(t) = R ∞ −∞ g dx is a smooth function satisfying (1) h(x) = 0 for x < a, (2) h(x) = 1 for b < x, and (3) 0 < h(x) < 1 for a < x < b. k (iii) Now given positive real numbers a < b, construct a smooth function ψa,b on R which is (1) identically one on the ball of radius a, (2) identically zero outside the ball of radius b, and (3) is strictly between 0 and 1 at intermediate points. (iv) For r > 0, let B(r) denote the open ball of radius r centered at the origin in Rk. Show that there exists a ˆ k finite open cover {Ui} (1 ≤ i ≤ m) of M by coordinate charts, with maps φi : Ui → Ui ⊂ R , chosen to have the following two properties: ˆ k • Ui ⊃ B(2), i.e. each Ui contains an open ball of radius 2 centered at the origin in R , −1 • the open subsets {φi (B(1))} cover M. k (v) Let ψ be the map ψ1,2 constructed in (iii). Verify that for each index i, the map fi : M → R defined below is smooth: ( ψ(φi(x)) · φi(x) x ∈ Ui fi = 0 x∈ / Ui. k (vi) Define the maps gi : M → R × R by gi(p) = (fi(p), ψi(p)), and show that the associated map G : M → k+1 m (R ) defined by p 7→ (g1(p), . , gm(p)) yields an injective immersion (and hence an embedding) of the manifold M into m(k + 1)-dimensional Euclidean space. 4) Projective spaces. Define the n-dimensional real projective space, denoted RP n, to be the quotient of Sn by the antipodal map p 7→ −p. (i) Show that RP n is a compact n-dimensional manifold (use either the intrinsic or extrinsic definition, see exercise (4) below). (ii) Viewing S1 ⊂ S2 as the equator, we have a natural embedding RP 1 ⊂ RP 2. Show that this submanifold cannot be realized as the pre-image f −1(r) of a regular value r of a smooth function f : RP 2 → R. [Hint: consider connected neighborhoods of RP 1 inside RP 2.] (iii) Show that the sphere bundle of the 2-sphere T 1S2 is in fact diffeomorphic to RP 3. (iv) Show that RP 2 embeds in R4, via the following steps: • show that RP 2 is diffeomorphic to the union of a Moebius band M and a 2-dimensional disk D2 along their common boundary. • show that there exists a 3-dimensional smooth curve γ ⊂ R3 which simultaneously arises as the boundary of an embedded D2, and as the boundary of an embedded M [Hint: find a convenient embedding of M first, then think of “flattening out” the boundary curve.] • the two embeddings of M and D2 into R3 might intersect each other; show that this can be fixed in R4..
Load more

Recommended publications
  • Connections on Bundles Md
    Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II.
    [Show full text]
  • FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject.
    [Show full text]
  • 3. Tensor Fields
    3.1 The tangent bundle. So far we have considered vectors and tensors at a point. We now wish to consider fields of vectors and tensors. The union of all tangent spaces is called the tangent bundle and denoted TM: TM = ∪x∈M TxM . (3.1.1) The tangent bundle can be given a natural manifold structure derived from the manifold structure of M. Let π : TM → M be the natural projection that associates a vector v ∈ TxM to the point x that it is attached to. Let (UA, ΦA) be an atlas on M. We construct an atlas on TM as follows. The domain of a −1 chart is π (UA) = ∪x∈UA TxM, i.e. it consists of all vectors attached to points that belong to UA. The local coordinates of a vector v are (x1,...,xn, v1,...,vn) where (x1,...,xn) are the coordinates of x and v1,...,vn are the components of the vector with respect to the coordinate basis (as in (2.2.7)). One can easily check that a smooth coordinate trasformation on M induces a smooth coordinate transformation on TM (the transformation of the vector components is given by (2.3.10), so if M is of class Cr, TM is of class Cr−1). In a similar way one defines the cotangent bundle ∗ ∗ T M = ∪x∈M Tx M , (3.1.2) the tensor bundles s s TMr = ∪x∈M TxMr (3.1.3) and the bundle of p-forms p p Λ M = ∪x∈M Λx . (3.1.4) 3.2 Vector and tensor fields.
    [Show full text]
  • WHAT IS a CONNECTION, and WHAT IS IT GOOD FOR? Contents 1. Introduction 2 2. the Search for a Good Directional Derivative 3 3. F
    WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG Abstract. In the study of differentiable manifolds, there are several different objects that go by the name of \connection". I will describe some of these objects, and show how they are related to each other. The motivation for many notions of a connection is the search for a sufficiently nice directional derivative, and this will be my starting point as well. The story will by necessity include many supporting characters from differential geometry, all of whom will receive a brief but hopefully sufficient introduction. I apologize for my ungrammatical title. Contents 1. Introduction 2 2. The search for a good directional derivative 3 3. Fiber bundles and Ehresmann connections 7 4. A quick word about curvature 10 5. Principal bundles and principal bundle connections 11 6. Associated bundles 14 7. Vector bundles and Koszul connections 15 8. The tangent bundle 18 References 19 Date: 26 March 2008. 1 1. Introduction In the study of differentiable manifolds, there are several different objects that go by the name of \connection", and this has been confusing me for some time now. One solution to this dilemma was to promise myself that I would some day present a talk about connections in the Olivetti Club at Cornell University. That day has come, and this document contains my notes for this talk. In the interests of brevity, I do not include too many technical details, and instead refer the reader to some lovely references. My main references were [2], [4], and [5].
    [Show full text]
  • Vector Bundles and Connections
    VECTOR BUNDLES AND CONNECTIONS WERNER BALLMANN The exposition of vector bundles and connections below is taken from my lecture notes on differential geometry at the University of Bonn. I included more material than I usually cover in my lectures. On the other hand, I completely deleted the discussion of “concrete examples”, so that a pinch of salt has to be added by the customer. Standard references for vector bundles and connections are [GHV] and [KN], where the interested reader finds a rather comprehensive discussion of the subject. I would like to thank Andreas Balser for pointing out some misprints. The exposition is still in a preliminary state. Suggestions are very welcome. Contents 1. Vector Bundles 2 1.1. Sections 4 1.2. Frames 5 1.3. Constructions 7 1.4. Pull Back 9 1.5. The Fundamental Lemma on Morphisms 10 2. Connections 12 2.1. Local Data 13 2.2. Induced Connections 15 2.3. Pull Back 16 3. Curvature 18 3.1. Parallel Translation and Curvature 21 4. Miscellanea 26 4.1. Metrics 26 4.2. Cocycles and Bundles 27 References 28 Date: December 1999. Last corrections March 2002. 1 2 WERNERBALLMANN 1. Vector Bundles A bundle is a triple (π,E,M), where π : E → M is a map. In other words, a bundle is nothing else but a map. The term bundle is used −1 when the emphasis is on the preimages Ep := π (p) of the points p ∈ M; we call Ep the fiber of π over p and p the base point of Ep.
    [Show full text]
  • On Manifolds of Negative Curvature, Geodesic Flow, and Ergodicity
    ON MANIFOLDS OF NEGATIVE CURVATURE, GEODESIC FLOW, AND ERGODICITY CLAIRE VALVA Abstract. We discuss geodesic flow on manifolds of negative sectional curva- ture. We find that geodesic flow is ergodic on the tangent bundle of a manifold, and present the proof for both n = 2 on surfaces and general n. Contents 1. Introduction 1 2. Riemannian Manifolds 1 2.1. Geodesic Flow 2 2.2. Horospheres and Horocycle Flows 3 2.3. Curvature 4 2.4. Jacobi Fields 4 3. Anosov Flows 4 4. Ergodicity 5 5. Surfaces of Negative Curvature 6 5.1. Fuchsian Groups and Hyperbolic Surfaces 6 6. Geodesic Flow on Hyperbolic Surfaces 7 7. The Ergodicity of Geodesic Flow on Compact Manifolds of Negative Sectional Curvature 8 7.1. Foliations and Absolute Continuity 9 7.2. Proof of Ergodicity 12 Acknowledgments 14 References 14 1. Introduction We want to understand the behavior of geodesic flow on a manifold M of constant negative curvature. If we consider a vector in the unit tangent bundle of M, where does that vector go (or not go) when translated along its unique geodesic path. In a sense, we will show that the vector goes \everywhere," or that the vector visits a full measure subset of T 1M. 2. Riemannian Manifolds We first introduce some of the initial definitions and concepts that allow us to understand Riemannian manifolds. Date: August 2019. 1 2 CLAIRE VALVA Definition 2.1. If M is a differentiable manifold and α :(−, ) ! M is a dif- ferentiable curve, where α(0) = p 2 M, then the tangent vector to the curve α at t = 0 is a function α0(0) : D ! R, where d(f ◦ α) α0(0)f = j dt t=0 for f 2 D, where D is the set of functions on M that are differentiable at p.
    [Show full text]
  • MATH534A, Problem Set 8, Due Nov 13
    MATH534A, Problem Set 8, due Nov 13 All problems are worth the same number of points. 1. Let M; N be smooth manifolds of the same dimension, F : M ! N be a smooth map, and let A ⊂ M have measure 0. Prove that F (A) has measure 0. 2. Let M be a smooth manifold, and let A ⊂ M have measure 0. Prove that A 6= M. 3. Let M be a smooth manifold of dimension m, and let π : TM ! M be the natural projection, −1 m m π(p; v) = p. For a chart (U; φ) on M, define φ^: π (U) ! R × R by φ^(p; v) = (φ(p); dpφ(v)): (a) Prove that there is a unique topology on TM such that for any chart (U; φ) on M the set −1 −1 2m π (U) ⊂ TM is open, and φ^ maps π (U) homeomorphically to an open subset of R . (b) Show that TM with this topology is Hausdorff and second countable. Hence show that TM endowed with charts (π−1(U); φ^) is a smooth manifold. 4. (a) Let F : M ! N be a smooth map. Consider the map dF : TM ! TN given by dF (p; v) = (F (p); dpF (v)). This map is sometimes called the global differential of F (but it is also okay to call it the differential of F ). Prove that this map is smooth. n n (b) Let F : M ! R be a smooth map. Consider the map TM ! R given by (p; v) 7! dpF (v). Prove that this map is smooth.
    [Show full text]
  • Arxiv:1505.02430V1 [Math.CT] 10 May 2015 Bundle Functors and Fibrations
    Bundle functors and fibrations Anders Kock Introduction The notions of bundle, and bundle functor, are useful and well exploited notions in topology and differential geometry, cf. e.g. [12], as well as in other branches of mathematics. The category theoretic set up relevant for these notions is that of fibred category, likewise a well exploited notion, but for certain considerations in the context of bundle functors, it can be carried further. In particular, we formalize and develop, in terms of fibred categories, some of the differential geometric con- structions: tangent- and cotangent bundles, (being examples of bundle functors, respectively star-bundle functors, as in [12]), as well as jet bundles (where the for- mulation of the functorality properties, in terms of fibered categories, is probably new). Part of the development in the present note was expounded in [11], and is repeated almost verbatim in the Sections 2 and 4 below. These sections may have interest as a piece of pure category theory, not referring to differential geometry. 1 Basics on Cartesian arrows We recall here some classical notions. arXiv:1505.02430v1 [math.CT] 10 May 2015 Let π : X → B be any functor. For α : A → B in B, and for objects X,Y ∈ X with π(X) = A and π(Y ) = B, let homα (X,Y) be the set of arrows h : X → Y in X with π(h) = α. The fibre over A ∈ B is the category, denoted XA, whose objects are the X ∈ X with π(X) = A, and whose arrows are arrows in X which by π map to 1A; such arrows are called vertical (over A).
    [Show full text]
  • Integral Formulas for a Foliation with a Unit Normal Vector Field
    mathematics Article Integral Formulas for a Foliation with a Unit Normal Vector Field Vladimir Rovenski Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel; [email protected] Abstract: In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation F and a unit vector field N orthogonal to F, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of F with respect to N and the curvature tensor of the induced connection on the distribution D = TF ⊕ span(N), and this decomposition of D can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation. Keywords: Riemannian manifold; foliation; harmonic distribution; Newton transformation; shape operator; r-th mean curvature MSC: 53C12; 53C17 1. Introduction Foliations, which are defined as partitions of a manifold into collections of submani- Citation: Rovenski, V. Integral folds of the same dimension, called leaves, appeared in the 1940s in the works of G. Reeb Formulas for a Foliation with a Unit and Ch. Ehresmann, culminating in the book [1]. Since then, the subject has enjoyed a Normal Vector Field. Mathematics rapid development, see e.g., [2]. Extrinsic geometry of foliations of Riemannian manifolds 2021, 9, 1764. https://doi.org/ 10.3390/math9151764 was developed in recent years, see surveys in [3–9], and has applications in differential geometry and analysis on real and complex manifolds.
    [Show full text]
  • Chapter 3 Connections
    Chapter 3 Connections Contents 3.1 Parallel transport . 69 3.2 Fiber bundles . 72 3.3 Vector bundles . 75 3.3.1 Three definitions . 75 3.3.2 Christoffel symbols . 80 3.3.3 Connection 1-forms . 82 3.3.4 Linearization of a section at a zero . 84 3.4 Principal bundles . 88 3.4.1 Definition . 88 3.4.2 Global connection 1-forms . 89 3.4.3 Frame bundles and linear connections . 91 3.1 The idea of parallel transport A connection is essentially a way of identifying the points in nearby fibers of a bundle. One can see the need for such a notion by considering the following question: Given a vector bundle π : E ! M, a section s : M ! E and a vector X 2 TxM, what is meant by the directional derivative ds(x)X? If we regard a section merely as a map between the manifolds M and E, then one answer to the question is provided by the tangent map T s : T M ! T E. But this ignores most of the structure that makes a vector 69 70 CHAPTER 3. CONNECTIONS bundle interesting. We prefer to think of sections as \vector valued" maps on M, which can be added and multiplied by scalars, and we'd like to think of the directional derivative ds(x)X as something which respects this linear structure. From this perspective the answer is ambiguous, unless E happens to be the trivial bundle M × Fm ! M. In that case, it makes sense to think of the section s simply as a map M ! Fm, and the directional derivative is then d s(γ(t)) − s(γ(0)) ds(x)X = s(γ(t)) = lim (3.1) dt t!0 t t=0 for any smooth path with γ_ (0) = X, thus defining a linear map m ds(x) : TxM ! Ex = F : If E ! M is a nontrivial bundle, (3.1) doesn't immediately make sense because s(γ(t)) and s(γ(0)) may be in different fibers and cannot be added.
    [Show full text]
  • On Fibre Bundles and Differential Geometry
    Lectures On Fibre Bundles and Differential Geometry By J.L. Koszul Notes by S. Ramanan No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the Tata Insti- tute of Fundamental Research, Apollo Pier Road, Bombay-1 Tata Institute of Fundamental Research, Bombay 1960 Contents 1 Differential Calculus 1 1.1 .............................. 1 1.2 Derivationlaws ...................... 2 1.3 Derivation laws in associated modules . 4 1.4 TheLiederivative... ..... ...... ..... .. 6 1.5 Lie derivation and exterior product . 8 1.6 Exterior differentiation . 8 1.7 Explicit formula for exterior differentiation . 10 1.8 Exterior differentiation and exterior product . 11 1.9 The curvature form . 12 1.10 Relations between different derivation laws . 16 1.11 Derivation law in C .................... 17 1.12 Connections in pseudo-Riemannian manifolds . 18 1.13 Formulae in local coordinates . 19 2 Differentiable Bundles 21 2.1 .............................. 21 2.2 Homomorphisms of bundles . 22 2.3 Trivial bundles . 23 2.4 Induced bundles . 24 2.5 Examples ......................... 25 2.6 Associated bundles . 26 2.7 Vector fields on manifolds . 29 2.8 Vector fields on differentiable principal bundles . 30 2.9 Projections vector fields . 31 iii iv Contents 3 Connections on Principal Bundles 35 3.1 .............................. 35 3.2 Horizontal vector fields . 37 3.3 Connection form . 39 3.4 Connection on Induced bundles . 39 3.5 Maurer-Cartan equations . 40 3.6 Curvatureforms. 43 3.7 Examples ......................... 46 4 Holonomy Groups 49 4.1 Integral paths . 49 4.2 Displacement along paths . 50 4.3 Holonomygroup ....................
    [Show full text]
  • Some Aspects of the Geodesic Flow
    Some aspects of the Geodesic flow Pablo C. Abstract This is a presentation for Yael’s course on Symplectic Geometry. We discuss here the context in which the geodesic flow can be understood using techniques from Symplectic Geometry. 1 Introduction We begin defining our object of study. Definition 1.1. Given a closed manifold M with Riemannian metric g, the co- geodesic flow is defined as the Hamiltonian flow φt on the cotangent bundle (T ∗M,Θ)(Θ will always denote the canonical form on T ∗M) where the cor- 1 responding lagrangian is L = 2 g Suppose that the Legendre transform is given by (x; v) 7! (x; p). Then we know • H = hp; vi − L(x; v) • p is given as the fiber derivative of L. Namely, consider f(v) = L(x; v) for x fixed and then p(x,v) is given by p = Dvf 1 In our very particular case, we can easily calculate the transform using Eüler’s lemma for homogeneous functions (L is just a quadratic form here); ge wet hp; vi = hDvf ; vi = 2f(v) = 2L = g(v; v) This implies • the Legendre transform identifies the fibers of TM and T ∗M using the metric (v 7! g(v; ·)). • H = 2L − L = L. This shows that the co-geodesic flow induces a flow (which we will continue denoting φt) on TM such that φt = (γ; γ_ ) and γ satisfies the variational equation for the action defined by L = H. 1 d @L We have that dt @v = 0, so the norm of γ_ is constant.
    [Show full text]