Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds

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1 MA 755 Fall 05. Notes #1. I. Kogan. Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds Definition 1 An n-dimensional C k -differentiable manifold is a second countable, Hausdorff topological space together with a differentiable structure, given by a countable collection of open sets U α M (where α A N) and homeomorphisms ψ α : U α V α, where V α is an open connected subset of R n, with the following conditions are satisfied: α U α = M; For every U α and U β such that U α U β, let W α = ψ α (U α U β ) V α and let W β = ψ β (U α U β ) V β. Then the bijection ψ β ψα 1 : W α W β and its inverse are C k differentiable. Open set U α are called (coordinate) charts, maps ψ α are called coordinate maps. Their collection {(U α, ψ α )} is called atlas on M, or the differentiable structure on M, maps ψ β ψα 1 : W α W β are called transition maps. From now on we will assume, unless it is explicitly stated otherwise, that k = in the above definition, that is, the transition maps, are smooth. Such manifolds are called smooth manifolds. If the transition maps are analytic, then the manifold is called analytic. Problem 1 Cover a circle S 1 of radius one centered at the origin in R 2 with two coordinate charts: U 1 = S 1 {(0, 1)} and U 2 = S 1 {(0, 1)}. Write down explicitly (in coordinates) projections π 1 from U 1 to the line y = 1 and π 2 from U 2 to the line y = 1. Describe the sets W 1 = π 1 (U 1 U 2 ) and W 2 = π 1 (U 1 U 2 ) and compute the transition maps π 2 π1 1 : W 1 W 2 explicitly. Explain to yourself why π 1 and π 2 are coordinate maps. Problem 2 Provide similar covering for the the sphere S 2 R 3 with two coordinate charts, find coordinate maps that correspond to projections from the north and south polls. Find transition maps. Can you generalize this construction to an n-dimensional sphere S n R n+1? As always in math, once a category of objects is defined (think of a linear space, a topological space, a group, for example), we need to define what classes of objects inside this category are considered to be the same (think of terms: isomorphic, homeomorphic etc) and what class of maps between the objects it makes sense to consider (think of linear maps, continuous maps, group homomorphisms etc). Definition 2 Two smooth n-dimensional manifolds M, {(U α, ψ α )} and N, {(W β, ξ β )} are diffeomorphic if there is a bijection f: M N such that a M and U α a and W β f(a), the bijection ξ β f ψα 1 and its inverse are smooth maps. Problem 3 Illustrate what is going on by a picture 1

2 Two diffeomorphic manifolds considered to be the same in the manifold theory. Assume that M = N as sets and f is the identity map in the above definition. Does it mean that M, {(U α, ψ α )} and M, {(W β, ξ β )} are diffeomorphic? In other words, is it true that the same topological space can have essentially unique differential structure? The answer is no. More details are on p. 39 of Spivak, Differential Geometry v1. In particular it is proved that there is unique differential structure on R n for n 4, but for n = 4 it is unsettled. There is unique differential structure on S n (sphere of dimension n) for n 6, but there are 28 different diff. structures for n = 7 and more than 16 million for n = 31. If M, {(U α, ψ α )} and M, {(W β, ξ β )} are diffeomorphic, then the corresponding atlases are called compatible. The union of all charts of all atlases compatible with M, {(U α, ψ α )} is called the universal atlas of M, {(U α, ψ α )}. Note that the universal atlas may contain uncountable number of charts. Definition 3 A map f from a manifold M, {(U α, ψ α )} to a manifold N, {(W β, ξ β )}, where dim M = m and dim N = n, is smooth if a M and U α a and W β f(a) the map ξ β f ψα 1 : R m R n is smooth. Problem 4 Explain why, for a given a, checking smoothness using a single chart on M containing a and a single chart on N containing containing f(a), instead of all such charts is sufficient. Definition 4 An immersed submanifold of M is a manifold M together with an injective smooth map ι: M M. Definition 5 An submanifold ι: M M is embedded, or regular,if for every point p ι(m) there exists an open nighborhood of Ũ M such that ι 1 (Ũ ι(m)) is a connected open subset of M. Theorem 6 Assume that a manifold M is a subset of M and the inclusion map ι is an embedding. Then near every a M there exists chart Ũ M, with coordinate functions x 1,..., x m such that Ũ M is given by {x x n+1 = 0,..., x m = 0}. Such coordinate system is called slice coordinates. See more on p.15 of the text-book. Problem 5 A two dimensional sphere S 2 = {(x, y, z) x 2 +y 2 +z 2 = 1} is embedded in in R 3. Find slice coordinates on the upper subset {(x, y, z) z > 0 R 3 }. Example 7 Figure eight, defined by a map φ(t) = (sin(2 arctan t), 2 sin(4 arctan t)), is not an embedded but an immersed submanifold of R 2. The same image is obtained by However, the map φ φ 1 : R R is not smooth. φ(t) = ( sin(2 arctan t), 2 sin(4 arctan t)) 2

3 2 Tangent spaces on a manifold. Definition 8 A function f: M R is smooth if a M and for all charts (U α, ψ α ) such that U α a the map f ψα 1 : R n R is smooth. Definition 9 A smooth curve is a smooth mapping γ: R M. More precisely if U α is any coordinate chart which contains a point γ(t) for some t R then the map ψ α γ: R R n is smooth. Problem 6 Consider a sphere S 2 = {(x, y, z) x 2 + y 2 + z 2 = 1} R 3. Let γ(t) = (x(t), y(t), z(t)) be a curve on S 2. What equation does the tangent to the curve γ(t) = (x (t), y (t), z (t)) satisfy? S 2 is a two dimensional manifold, so it is more appropriate to use two coordinate functions. Use a coordinate chart U 1 which excludes north pole, find explicit projection π 1 : U 1 R 2 and compute equations of the curve γ = π 1 γ. Write explicitly the tangent vector γ. Definition 10 Two curves γ 1 : R M and γ 2 : R M, such that γ 1 (0) = γ 2 (0) = a are said to have the same tangent at a (or in other words to have the same first order contact at a) if for any U α a curves φ α γ 1 : R R n and φ α γ 2 : R R n have the same tangent at the point φ α (a). Problem 7 In the context of Problem 6 show that two curves on the sphere S 2 R 3 have the same tangent at a point a, according to Definition 10 if and only if they have the same tangent in the usual sense as curves in R 3. Problem 8 Note that a curve is defined to be a map R M, and not just as the image of this map. This exercise underscores that the image of a smooth curve can have cusps and self intersection, that two curves with the same image have different tangents at the same point. In other words parametrization matters! 1. Sketch the curves γ(t) = (x(t), y(t)) = (t 2, t 3 ), and β(t) = (x(t), y(t)) = (t 2 1, t 3 t) on R 2. Are these curves smooth in the sense of definition 9? Does their images on R 2 look smooth? Compute vectors γ(0) and β(0). To which point of R 2 are these tangent vector attached? 2. Sketch the curves β 1 (t) = (x(t), y(t)) = (4t 2 1, 8t 3 2t) = ((2t) 2 1, (2t) 3 (2t)) on R 2, compare it with the curve defined by β. Note that β and β 1 are different curves according to definition 9. Compute β 1 (0). To which point of R 2 is this tangent vector attached? Compare it with β(0). 3. Sketch the curves β 2 (t) = (x(t), y(t)) = ( (t 1) 2 1, (t 1) 3 (t 1) ) = (t 2 2t, (t 1)(t 2 2t)) and β 3 (t) = (x(t), y(t)) = ( (t + 1) 2 1, (t + 1) 3 (t + 1) ) on R 2, compare them with the curve defined by β. Compute β 2 (0) and β 3 (0). To which point of R 2 are these tangent vectors attached? Definition 11 The tangent space at a point a M (denoted T M a ) is a set of equivalence classes of curves, γ: R M such that γ(0) = a. Two curves are in the same class if they have the same tangent at a. 3

4 Problem 9 Define addition and scalar multiplication on the elements of T M a. Show that T M a is isomorphic to R n as a linear space. T M a can be imagined as a linear space R n of vectors attached to the point a. Note that one can not add vectors attached to two different points. A smooth map F between two manifolds induces smooth maps between tangent spaces, called it differential of F. 3 Directional derivatives and derivations For every smooth curve γ: R M and a smooth function f: M R the function f γ: R R is smooth. Definition 12 The derivative of a function f in the direction of vector γ T M a at a point a = γ(0) is defined by γf(a) = d f(γ(t)). It is not difficult to check that γ satisfies the following two important properties, f, g C (M) smooth functions on M and c R: γ(f + cg)(a) = γf(a) + c γg(a) linarity, (1) γ(fg)(a) = ( γf(a)) g(a) + f(a) γg(a) Leibniz (product) rule. (2) An assignment of a tangent vector at every point a M, that varies smoothly from point to point is called a a vector field on M (see Definition 16). A map C (M) C (M), that satisfies the above two properties is called a smooth derivation. There is a one-to-one correspondence between vector fields on a manifold and derivations explored in MA 555 notes 2. Remark 1 We often omit coordinate maps and identify a point a in M with its coordinates: p = (x 1,..., x m ). The corresponding basis for T M a consists of vectors E 1,..., E m, such that E i (x j ) = δ j i. It is natural to denote this basis as,... x 1 x (or simply m 1,..., m ). A tangent vectors X T M a is written as X = ξ ξ n x 1 x. One need to be careful, however, with n what happens under the change of coordinates!!! Similarly we can identify a function f : M R with its coordinate presentation f(x 1,..., x m ). Then Xf(a) = ξ 1 f (a) +... ξ n f x 1 x (a) n Problem 10 a) Let γ(t) = (sin t cos t, cos 2 t, sin t) be a curve on the unit sphere S 2 R 3. Let a = (0, 1, 0) S 2 and f : S 2 R defined by f(x, y, z) = x + y + z. Compute γf(a). b) Use the map π 1 obtained in Exersice 6 to find a curve γ(t) = π 1 γ: R R 2 and the function f = f π 1 1 : R2 R. Compute b = π 1 (a) R 2 and γ f(b). Compare you answer with the answer in part a). Make a picture that relates parts a) and b) of the problem. 4 The differential of a map. Immersions and submersions. Definition 13 Let F : M N be a smooth map between manifolds. The differential of F at a is a map F a : T M a T M F (a), defined as follows: 4

5 Assume that γ T M a is defined by a smooth curve γ: R M, such that γ(0) = a. Then γ = F γ is a smooth curve on N, such that γ(0) = F (a). We define F a ( γ) = γ. In local coordinates F a is given by the Jacobian of F at a. Vector F a ( γ) is called the pushforward of γ under F. Assume dim M = m and dim N = n then If rank F a = m at every point then F is called local immersion. If F is also one-to-one it is called immersion If rank F a = n at every point then F is called submersion If rank F a = n = m at every point then F is called local diffeomorphism. one-to-one, then it is a diffeomorphism. If F is also 5 Tangent bundle Definition 14 A tangent bundle T M of a manifold M is the union of tangent spaces at all point of M, that is T M = a M T M a = {a, γ(a)} Theorem 15 If M is smooth m-dimensional manifold, then T M is a smooth 2m-dimensional manifold. Moreover T M has a differential structure, such that the surjection π: T M M, given by π(a, γ(a)) = a, is smooth. proof: Main idea: one can show that if {U α, φ α } is an atlas on M then { a Uα T M a, (φ α, φ α )} is an atlas on T M. Definition 16 A smooth vector field on M is a smooth map X: M T M such that π X = id. Example 17 Let x 1,..., x n be the standard Cartesian coordinate functions on R n. (Note that that x i : R n R i = 1..n are smooth functions.) Given a smooth curve γ(t) = (x 1 (t),..., x n (t)) one ) obtains a vector γ(0) =,..., dxn attached to a point a = γ(0). The components of ( dx 1 this vector are real numbers. Thus if a R n then the tangent space T R n a = R n is a set of vectors attached to the point a. The tangent bundle T R n = R n R n = R 2n Problem 11 Find an atlas on the tangent bundle on T S 1, where S 1 is a unit circle. A tangent bundle is an example of a vector bundle. See p16 of the text-book for the precise definition. A vector field is a section of this bundle. Both T R n = R n R n and T S 1 = S 1 R are trivial. T S 2 is an example of non-trivial bundle (it is not diffeomorphic to S 2 R 2 ). 6 Tangent vectors under a change of coordinates Let (U 1, φ 1 ) and (U 2, φ 2 ) be two overlapping charts on an n-dimensional manifold M. Then φ 2 φ 1 1 is an invertible smooth map from an open subset V R n to an open subset W R n. Let x 1,..., x n 5

6 be coordinate functions on the first copy of R n and y 1,..., y n be coordinate functions on the first copy of R n. Then φ 2 φ 1 1 defines an invertible differentiable map φ: V W : y 1 = y 1 (x 1,..., x n )... (3) y n = y n (x 1,..., x n ) Let γ(t) be curve on M, such that γ(0) = a U 1 U 2. Let φ 1 (a) = x and φ 2 (a) = y. Then the curves x(t) = φ 1 γ(t) and y(t) = φ 2 γ(t) define the coordinates of the tangent vector γ. Indeed ( it has coordinates ẋ = dx 1 ) T,..., dxn (we arrange the coordinates in a column vector) in ( ) the first coordinate system, and coordinates ẏ = dy 1 T,..., dyn in the second. Theorem 18 ẏ = J φ (x)ẋ. (4) proof: We have y(t) = ( y 1 (x 1 (t),..., x n (t)),..., y n (x 1 (t),..., x n (t)) ). Hence ( n y 1 dxi ẏ(t) = (x) xi,..., i=1 n y n dxi (x) xi i=1 ) Corollary 19 y 1. y n (J φ ) T x 1. x n, (5) 7 Integral curves and flows of vector fields Definition 20 The integral curve of the vector field V through a point a R n on R n is a curve γ: R R n such that γ(0) = a and dγ = V (γ(t)). (6) Let x 1,... x n be coordinate functions on R n, then V = ξ ξ n x 1 x, where ξ i, n smooth functions on R n, and (6) becomes a system of the 1st order ODE s i = 1..n are dx 1 dx n = ξ 1 (x) (7). (8) = ξ n (x). (9) 6

7 A classical theorem assures that there exists a unique smooth solution of this system with any initial condition x i (0) = a i R, i = 1..n. Equivalently there exists a unique integral curve of the vector field V through every point. Due to the uniqueness, the integral curves do not intersect. Thus R n is a disjoint union of integral curves of a vector field V. We say that integral curves form a foliation of R n. Note that if V (a) is the zero vector, then the integral curve through a is a point: γ(t) = a, t R. Problem 12 Find the integral curve of the vector field V = x + x condition a) (x 0, y 0 ); b) (0, 1). y on R2 with the initial Definition 21 A smooth map Φ: R R n R n is called a flow of a vector field V if dφ (t, x) = V (Φ(t, x)). Φ(0, x) = x. The existence of such map follows from the existence of the integral curve through each point. Indeed, for every fixed point x 0 R n the map Φ(t, x 0 ): R R n is a smooth integral curve of V through x 0. By fixing arbitrary value t 0 R we obtain the smooth map Φ t0 (x) = Φ(t 0, x) : R n R n. Theorem 22 The set of maps Φ t, t R have the following properties: Φ 0 = id (that is, Φ 0 is the identity map: Φ 0 (x) = x for all x R n ). Φ t1 Φ t2 = Φ t1 +t 2. Φ t = Φ t 1. In particular, this means that every map Φ t has a smooth inverse and therefore is a diffeomorphism. Corollary 23 Let Φ(t, x) be the flow of a vector field V. The set of maps {Φ t : R n R n } is a one-parameter group of diffeomorphism isomorphic to the abelian group R. Problem 13 For vector field V = x + x y on R2 compute the flow Φ(t, x). Check that the properties of Theorem 22 are satisfied. Compute the inverse of the diffeomorphism Φ 2. 8 Lie series and Lie derivatives of a function. Theorem 24 Let f: R n R be an analytic function and V be an analytic vector field on R n, whose flow is Φ(t, x). Then for all x R n we have the following power series expansion along the integral curve γ(t) = Φ(t, x), called the Lie series of f. f (Φ(t, x)) = f(γ(t)) = f(x) + tv f(x) + t2 2 V (V (f)) (x) +... = k=0 t k k! V k (f)(x) := e tv f(x). (10) 7

8 By V k f(x) we mean that the derivation V is applied k times to the function f, and the resulting function V k f is then evaluated at a point x. For a fixed point x, the result is a number. The last equality is just a natural abbreviation, motivated by Taylor expansion for the usual exponent function. e tv t k := k! V k. k=0 If we also introduce notation e tv x 0 = Φ(t, x 0 ), then the Lie series can be written in a compact form: f(e tv x) = e tv f(x). An outline of the proof of theorem 22. Start with the usual Taylor series: f (Φ(t, x)) = f(γ(t)) = f(γ(0))+ d ( ) f(γ(t)) + t2 d 2 f(γ(t)) +... = 2 k=0 t k ( ) d k f(γ(t)). k! d From the definition of the flow it follows that f(γ(t)) = d f(φ(t, x)) = V f(x). Use this ) k as the basis case to prove by induction that f(γ(t)) = V k f(x) ( d From (10) it follows that f(e tv x) f(x) lim = V f(x). t 0 t Similar limits along the integral curves of V can be computed not only for functions, but also for vector fields, differential forms and other tensor fields. Such limits are called Lie derivatives. Thus the Lie derivative of f along the flow of V is simply the directional derivative V f. Problem 14 a) Given a vector field V = x + x y on R2, find the Lie derivatives and Lie series for f(x, y) = x and g(x, y) = y. Compare the results with what you found in Problem 12. Also find Lie derivatives and Lie series for f(x, y) = x 2 + y 2 and g(x, y) = y x2 2. Which one of these two functions is constant along any integral curve? Why? b) Given a vector field V = y x +x y on R2, find the Lie derivatives and Lie series for f(x, y) = x and g(x, y) = y. Could you find/guess a function that is constant along any integral curve? 8