Tangent bundles, vector fields

Transcription

1 Location Department of Mathematical Sciences,, G Main Reference: [Lee]: J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer-Verlag, Homepage for the book, where you may find a list of corrections: Supplementary reading W. Boothby, An Introduction to Differentiable Manifolds & Riemannian Geometry, Second Edition, Academic Press, L. Conlon, Differentiable Manifolds. A First Course, Birkhauser, Tangent bundles, vector fields Mon., , 9 12 For manifolds M R n, the tangent vectors to all curves on M form a subset TM M R n, such that T p M = TM ({p} R n ) is the tangent space at p for every p M. For abstract manifolds, one can extend this construction collecting all tangent vectors on M to yield the tangent bundle π : TM M; its total space TM is given the structure of a smooth manifold, and T p M = π 1 (p) is the tangent space at p a vector space. The strength of the construction relies on the combination of the (global) smooth structure with the (local) vector space structure allowing to perform vector space operations (linear combinations) which are not possible on the manifold itself. A vector field associates to every point a tangent vector at that point in a continuous or smooth way. In the abstract setting, a vector field on M is a section of the bundle map π, i.e., a continuous/smooth map Y : M TM such that π Y = Id M. A vector field on M defines both a derivation on function space Y : C (M) C (M), f Y f, and a dynamical system on M generalizing the concept of a system of differential equations in Euclidean space.

2 [Lee] Ch. 4, pp (from [Lee]) Probl. 3-4, p. 79 Ex. 4.1, p. 82 Probl. 4-1, p. 100 Operations on vector fields Mon., , 12:30 15 Aims and Content The main use of a vector field stems from an interpretation as a dynamical system: A vector field associates to every point a direction exactly as in the case of a system of differential equations in Euclidean space R n. A point on the manifold corresponds to an initial condition, and the integral curve through that point corresponds to a solution of the system with given initial condition. Together, the solutions through all points define the flow of the vector field. Using the point of view that vector fields define derivations, one can apply vector fields one on top of the other (like mixed partial derivatives). It is an important insight that, unlike for mixed partial derivatives, the outcome may depend on the order of the vector fields; and that the difference [V,W] : C (M) C (M), [V,W]f = V Wf WV f defines a new vector field [V, W]f. This Lie-bracket operation gives important information on the geometry of the manifold (cf. below). To exploit it, we investigate formal properties of the Lie bracket: bilinearity, antisymmetry, the Jacobi identity (a particular form of non-associativity!) and naturality. The geometric interpretation of the Lie-bracket is in terms of Lie-derivatives of vector fields using flows. We explain the ideology behind Lie-derivatives and why Lie-brackets can be used to decide whether two vector fields (resp. their flows) commute. [Lee] Ch. 4, pp ; Ch. 17, pp (highlights), Ch. 18, pp (highlights). (from [Lee]) Problems 4-5, 4-6, 4-11(b), p. 101.

3 The inverse function theorem and its relatives Wed., , 9 12 The differential of a smooth map between Euclidean spaces is (by definition) the best approximating linear map. Which properties are preserved by the linearisation? This is the theme of the inverse function theorem (IFT), a crucial result from mathematical analysis that allows to propagate knowledge about properties of the differential of a smooth map to properties of the map itself (at least locally). In particular, if the differential is a bijection, then the original map defines local diffeomorphisms (in general not global ones). We will not go through the proof of this theorem (this is done in another phd-course); instead we shall explain, apply and modify this theorem. First of all, there is a generalization of the theorem concerning smooth maps between manifolds and their differentials, viewed as a collection of linear maps. A bit more general, we investigate maps of constant rank (i.e., the differential has the same rank at every point). Such a map is locally conjugate to a projection map. When the differential is injective at all points, the map is an immersion. The most important case is a submersion with a differential that is surjective, i.e., of maximal rank, at every point. Interesting applications will be explained during the afternoon session. [Lee] Ch. 7, pp The proof of the IFT (pp ) will not be presented. Several of you have seen it during the course on Analysis and Topology. A map F : RP 2 R 4 is defined by F([x,y,z]) = 1 x 2 +y 2 +z 2 (x 2 y 2,xy,xz,yz). Show that F is well-defined (i.e., if [x 1,y 1,z 1 ] = [x 2,y 2,z 2 ], then the expression above yields the same point in R 4 ) F is injective. (From above, we know that it suffices to consider the case (x,y,z) S 2 ) F is an immersion. Use Proposition 7.4 to conclude, that F is an embedding. Problem 7-5 [Hint: M compact (cf. pp. 552/553) implies: Every continuous function f : M R attains a maximal value on M at a certain point x M (and also a minimal value). What does this imply for the gradient f at x in local coordinates? And for the differential DF(x) with row vectors F i?]

4 When are level sets (sub)-manifolds? Wed., , 12:30 15 Many important manifolds are defined as solutions of equations: Among all n n-matrices, the orthogonal ones forming the Lie group O(n) are singled out as solutions of the matrix equation AA T = I. Under which conditions do the solutions of one (or several) equations form a smooth manifold, in fact a submanifold? Said more formally: Let Φ : M N denote a smooth map and let c N. The level set Φ 1 (c) consists of all solutions of the equation Φ(x) = c with x M. Under the crucial condition that c is a regular value, i.e., that the differential Φ is surjective for all solutions x, this level set is in fact a closed embedded submanifold whose dimension depends on the dimensions of the domain and codomain and the rank of the map. This follows quite easily from the constant rank theorem of the morning session. Important examples of manifolds arising as level sets are the spheres, the spaces of matrices of a fixed rank and several Lie groups: the orthogonal group O(n), the special linear and orthogonal groups SL(n,R), resp. SO(n) etc. [Lee], Ch. 8, pp ; pp Exercise 8.3, p. 183 Problem 8-1, p. 201 Problem 8-2, p. 201 (It is instructive to plot various level sets F 1 (c) around c = 0 and c = 1 1. What is special for c =? Determine F 1 ( 1 ).)

5 Tensors and tensor fields Fri., , 9 12 How can one define a metric on an abstract smooth manifold? In Euclidean space, the metric is derived from the scalar product. The length of a curve can then be defined via the length of velocity vectors along a parametrization of the curve. What you need in general is a way to measure lengths of tangent vectors. On an abstract smooth manifold, there is no obvious way of measuring lengths (or angles) on tangent vectors; this is an extra structure that one has to supply as a Riemannian metric. Metrics, measurements of areas, volumes, curvature etc. can be derived from tensor fields on the tangent bundle. To formulate the concept, we need to go into the tensor concept from linear algebra. Tensors can be seen as a smart way of looking at multilinear functions on a vector space (R n ). Typical examples are the usual scalar product (a 2-tensor) and the determinant (an n-tensor on a vector space of dimension n. In full generality, one can investigate tensors that are covariant in some and contravariant in other variables. Emphasis will be on concepts rather than proofs. To make the algebra of tensors applicable on manifolds, we need to introduce tensor bundles (consisting of tensors at each fibre; e.g., all possible scalar products on a given tangent space) and tensor fields that pick up a particular tensor (e.g., a particular scalar product) at each point in a consistent smooth manner. [Lee] Ch.11, p Problem 8-9, p. 201 (Plots for various values a are instructive) Problem 8-10, p. 202 (Hint: Try a linear non-surjective map between real vector spaces, or the map from Problem 8-2 from the last session) Exercise 11.3, p. 267 Exercise 11.4, p. 268

6 Towards Riemannian metrics Fri., , 12:30 15 Tensor fields can be pulled pack along smooth maps between two smooth manifolds making comparisons possible. A Riemannian metric on a manifold is a smooth symmetric (invariant under interchange of the vector variables) tensor field that, moreover, is positive definite at each point all requirements that generalize those of the scalar product. In local coordinates, it is given by a function with values given as symmetric positive definite n n-matrix g ij (x). From a Riemannian metric, one derives easily the length of a tangent vector and the angle between two such. From lengths of tangent vectors one can derive the length of a curve by integrating velocity vectors 1. The Riemannian distance between two points is then defined as the lowest upper bound of all curve lengths among curves connecting these two points (in favourable cases, it is the lenght of the shortest curves between the points). It is often difficult to calculate that distance, but it certainly defines a metric on the manifold (now as a metric space) with the same topology (open sets) as the topology given on the manifold via coordinate charts. Is there always a Riemannian metric on a given manifold? Yes! Locally defined scalar products can be pieced together via a partition of unity. [Lee] Ch. 11, pp , , , 284. Problem 11-10, p. 287 Your assessment of the Ph.D.-course: Was the topic interesting/relevant? How about the textbook? the lectures? the exercises? 1 A Riemannian metric also determines a volume form by a sort of determinant, allowing to calculate volumes of manifold pieces and the entire manifold by multiple integration; we will not have time to deal with this aspect

7 What would you recommend to be changed if the course were given another time? What should remain as it was this time?