CHAPTER 1 : DIFFERENTIABLE MANIFOLDS. 1.1 The definition of a differentiable manifold

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1 CHAPTER 1 : DIFFERENTIABLE MANIFOLDS 1.1 The efinition of a ifferentiable manifol Let M be a topological space. This means that we have a family Ω of open sets efine on M. These satisfy (1), M Ω (2) the union of any family of open sets is open (3) the intersection of a finite family of open sets is open We normally assume also the Hausorff property: For any pair x, y of istinct points there is a pair of nonoverlapping open sets U, V such that x U an y V. In any topological space one can efine the notion of convergence. A sequence x 1, x 2, x 3,... converges towars x M if any open set U such that x U contains all the points x n except a finite set. The basic example of a topological space is R n equippe with the Eucliean norm x 2 = x x x 2 n for x = (x 1,..., x n ). A set U R n is open if for any x U there is a positive number ɛ = ɛ(x) such that y U if x y < ɛ. The convergence is then the usual one: x (n) x if for any ɛ > 0 there is an integer n ɛ such that x x (n) < ɛ for n > n ɛ. Actually, all the spaces we stuy in (finite imensional) ifferential geometry are locally homeomorphic to R n. Definition. A topological space M is calle a smooth manifol of imension n if 1) there is a family of open sets U α (with α Λ) such that the union of all U α s is equal to M, 2) for each α there is a homeomorphism φ α : U α V α R n such that 3) the coorinate transformations φ α φ 1 β smooth functions in R n. on their omains of efinition are Example 1 R n is a smooth manifol. We nee only one coorinate chart U = M with φ : U R n the ientity mapping. Example 2 The same as above, but take M R n any open set. Example 3 Take M = S 1, the unit circle. Set U equal to the subset parametrize by the polar angle 0.1 < φ < π+0.1 an V equal to the set π < φ < 2π. Then U V consists of two intervals π < φ < π an 0.1 < φ < 0 2π 0.1 < φ < 2π. 1

2 2 The coorinate transformation is the ientity map φ φ on the former an the translation φ φ + 2π on the latter interval. Exercise Define a manifol structure on the unit sphere S 2. Example 4 The group GL(n, R) of invertible real n n matrices is a smooth manifol as an open subset of R n2. It is an open subset since it is a complement of the close surface etermine by the polynomial equation eta = Differentiable maps Let M, N be a pair of smooth manifols (of imensions m, n) an f : M N a continuous map. If (U, φ) is a local coorinate chart on M an (V, ψ) a coorinate chart on N then we have a map ψ f φ 1 from some open subset of R m to an open subset of R n. If the composite map is smooth for any pair of coorinate charts we say that f is smooth. The reaer shoul convince himself that the conition of smoothness for f oes not epen on the choice of coorinate charts. From elementary results in ifferential calculus it follows that if g : N P is another smooth map then also g f : M P is smooth. Note that we can write the map ψ f φ 1 as y = (y 1,..., y n ) = (y 1 (x 1,..., x m ),......, y n (x 1,..., x m )) in terms of the Cartesian coorinates. Smoothness of f simply means that the coorinate functions y i (x 1, x 2,..., x m ) are smooth functions. Remark In a given topological space M one can often construct ifferent inequivalent smooth structures. That is, one might be able to construct atlases {(U α, φ α )} an {(V α, ψ α )} such that both efine a structure of smooth manifol, say M U an M V, but the ientity map i : M U M V woul not be smooth. A famous example of this phenomen are the spheres S 7, S 11 (John Milnor, 1956). On the sphere S 7 there are exactly 28 inequivalent ifferentiable structures! On the Eucliean space R 4 there is an infinite number of ifferentiable structures (S.K. Donalson, 1983). A iffeomorphism is a one-to-one smooth map f : M N such that its inverse f 1 : N M is also smooth. The set of iffeomorphisms M M forms a group Diff(M). A smooth map f : M N is an immersion if at each point p M the rank of the erivative h x is equal to the imension of M. Here h = ψ f φ 1 with

3 the notation as before. Finally f : M N is an embeing if f is injective an it is an immersion; in that case f(m) N is an embee submanifol. A smooth curve on a manifol M is a smooth map γ from an open interval of the real axes to M. Let p M an (U, φ) a coorinate chart with p U. Assume that curves γ 1, γ 2 go through p, let us say p = γ i (0). We say that the curves are equivalent at p, γ 1 γ 2, if t φ(γ 1(t)) t=0 = t φ(γ 2(t)) t=0. This relation oes not epen on the choice of (U, φ) as is easily seen by the help of the chain rule: t ψ(γ 1(t)) t ψ(γ 2(t)) = (ψ φ 1 ) ( t φ(γ 1(t)) ) t φ(γ 2(t)) = 0 at the point t = 0. Clearly if γ 1 γ 2 an γ 2 γ 3 at the point p then also γ 1 γ 3 an γ 2 γ 1. Trivially γ γ for any curve γ through p so that is an equivalence relation. A tangent vector v at a point p is an equivalence class of smooth curves [γ] through p. For a given chart (U, φ) at p the equivalence classes are parametrize by the vector t φ(γ(t)) t=0 R n. Thus the space T p M of tangent vectors v = [γ] inherits the natural linear structure of R n. Again, it is a simple exercise using the chain rule that the linear structure oes not epen on the choice of the coorinate chart. We enote by T M the isjoint union of all the tangent spaces T p M. This is calle the tangent bunle of M. We shall efine a smooth structure on T M. Let p M an (U, φ) a coorinate chart at p. Let π : T M M the natural projection, (p, v) p. Define φ : π 1 (U) R n R n as φ(p, [γ]) = (φ(p), t φ(γ(t)) t=0). 3 If now (V, ψ) is another coorinate chart at p then ( φ ψ 1 )(x, v) = (φ(ψ 1 (x)), (φ ψ 1 ) (x)v), by the chain rule. It follows that φ ψ 1 is smooth in its omain of efinition an thus the pairs (π 1 (U), φ) form an atlas on T M, giving T M a smooth structure.

4 4 Example 1 If M is an open set in R n then T M = M R n. Example 2 Let M = S 1. Writing z S 1 as a complex number of unit moulus, consier curves through z written as γ(t) = ze ivt with v R. This gives in fact a parametrization for the equivalence classes [γ] as vectors in R. The tangent spaces at ifferent points z 1, z 2 are relate by the phase shift z 1 z 1 2 an it follows that T M is simply the prouct S 1 R. Example 3 In general, T M M R n. The simplest example for this is the unit sphere M = S 2. Using the spherical coorinates, for example, one can ientify the tangent space at a given point (θ, φ) as the plane R 2. However, there is no natural way to ientify the tangent spaces at ifferent points on the sphere; the sphere is not parallelizable. This is the content of the famous hairy ball theorem. Any smooth vector fiel on the sphere has zeros. (If there were a globally nonzero vector fiel on S 2 we woul obtain a basis in all the tangent spaces by taking a (oriente) unit normal vector fiel to the given vector fiel. Together they woul form a basis in the tangent spaces an coul be use for ientifying the tangent spaces as a stanar R 2.) Exercise The unit 3-sphere S 3 can be thought of complex unitary 2 2 matrices with eterminant =1. T S 3 = S 3 R 3. Use this fact to show that the tangent bunle is trivial, Let f : M N be a smooth map. We efine a linear map T p f : T p M T f(p) N, as T p f [γ] = [f γ], where γ is a curve through the point p. This map is expresse in terms of local coorinates as follows. Let (U, φ) be a coorinate chart at p an (V, ψ) a chart at f(p) N. Then the coorinates for [γ] T p M are v = t φ(γ(t)) t=0 an the coorinates for [f γ] T f(p) N are w = t ψ(f(γ(t))) t=0. But by the chain rule, w = (ψ f φ 1 ) (x) t φ(γ(t)) t=0 = (ψ f φ 1 ) (x) v with x = φ(p). Thus in local coorinates the linear map T p f is the erivative of ψ f φ 1 at the point x. Putting together all the maps T p f we obtain a map T f : T M T N.

5 5 Proposition. The map T f : T M T N is smooth. Proof. Recall that the coorinate charts (U, φ), (V, ψ) on M, N, respectivly, lea to coorinate charts (π 1 (U), φ) an (π 1 (V ), ψ) on T M, T N. Now ( ψ T f φ 1 )(x, v) = ((ψ f φ 1 )(x), (ψ f φ 1 ) (x)v) for (x, v) φ(π 1 (U)) R m R m. Both component functions are smooth an thus T f is smooth by efinition. If f : M N an g : N P are smooth maps then g f : M P is smooth an T (g f) = T g T f. To see this, the curve γ through p M is first mappe to f γ through f(p) N an further, by T g, to the curve g f γ through g(f(p)) P. In terms of local coorinates x i at p, y i at f(p) an z i at g(f(p)) the chain rule becomes the stanar formula, z i x j = k z i y k y k x j. 1.3 Vector fiels We enote by C (M) the algebra of smooth real value functions on M. A erivation of the algebra C (M) is a linear map : C (M) C (M) such that (fg) = (f)g + f(g) for all f, g. Let v T p M an f C (M). Choose a curve γ through p representing v. Set v f = t f(γ(t)) t=0. Clearly v : C (M) R is linear. Furthermore, v (fg) = t f(γ(t)) t=0g(γ(0)) + f(γ(0)) t g(γ(t)) t=0 = (v f)g(p) + f(p)(v g). A vector fiel on a manifol M is a smooth istribution of tangent vectors on M, that is, a smooth map X : M T M such that X(p) T p M. From the

6 6 previous formula follows that a vector fiel efines a erivation of C (M); take above v = X(p) at each point p M an the right- han-sie efines a smooth function on M an the operation satisfies the Leibnitz rule. We enote by D 1 (M) the space of vector fiels on M. As we have seen, a vector fiel gives a linear map X : C (M) C (M) obeying the Leibnitz rule. Conversely, one can prove that any erivation of the algebra C (M) is represente by a vector fiel. One can evelope an algebraic approach to manifol theory. In that the commutative algebra A = C (M) plays a central role. Points in M correspon to maximal ieals in the algebra A. Namely, any point p efines the ieal I p A consisting of all functions which vanish at the point p. The action of a vector fiel on functions is given in terms of local coorinates x 1,..., x n as follows. If v = X(p) is represente by a curve γ then (X f)(p) = t f(γ(t)) t=0 = k f x k x k t (t = 0) k X k (x) f x k. Thus a vector fiel is locally represente by the vector value function (X 1 (x),..., X n (x)). In aition of being a real vector space, D 1 (M) is a left moule for the algebra C (M). This means that we have a linear left multiplication (f, X) fx. The value of fx at a point p is simply the vector f(p)x(p) T p M. As we have seen, in a coorinate system x i a vector fiel efines a erivation with local representation X = k X k x k. In a secon coorinate system x k we have a representation X = X k x k obtain the coorinate transformation rule. Using the chain rule for ifferentiation we X k(x ) = j x k x j X j (x), for x k = x k (x 1,..., x n ). We shall enote k = x k repeate inices, an we use Einstein s summation convention over Let X, Y D 1 (M). We efine a new erivation of C (M), the commutator [X, Y ] D 1 (M), by [X, Y ]f = X(Y f) Y (Xf).

7 7 We prove that this is inee a erivation of C (M). [X, Y ](fg) = X(Y (fg)) Y (X(fg)) = X(fY g + gy f) Y (fxg + gxf) = (Xf)(Y g) + fx(y g) + (Xg)(Y f) + gx(y f) (Y f)(xg) fy (Xg) (Y g)(xf) gy (Xf) = f[x, Y ]g + g[x, Y ]f. Writing X = X k k an Y = Y k k we obtain the coorinate expression [X, Y ] k = X j j Y k Y j j X k. Thus we may view D 1 (M) simply as the space of first orer linear partial ifferential operators on M with the orinary commutator of ifferential operators. The commutator [X, Y ] is also calle the Lie bracket on D 1 (M). It has the basic properties (1) [X, Y ] is linear in both arguments (2) [X, Y ] = [Y, X] (3) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0. The last property is calle the Jacobi ientity. A vector space equippe with a Lie bracket satisfying the properties above is calle a Lie algebra. Other examples of Lie algebras: Example 1 Any vector space with the bracket [X, Y ] = 0. Example 2 The angular momentum Lie algebra with basis L 1, L 2, L 3 an nonzero commutators [L 1, L 2 ] = L 3 + cyclically permute relations. Example 3 The space of n n matrices with the usual commutator of matrices, [X, Y ] = XY Y X. Exercise Check the relations [X, fy ] = f[x, Y ] + (Xf)Y, an [fx, Y ] = f[x, Y ] (Y f)x for X, Y D 1 (M) an f C (M). Let f : M N be a iffeomorphism an X D 1 (M). We can efine a vector fiel Y = f X on N by setting Y (q) = T p f X(p) for q = f(p). In terms of local coorinates, y k Y = Y k = X j. y k x j y k

8 8 In the case M = N this gives back the coorinate transformation rule for vector fiels. Let X D 1 (M). Consier the ifferential equation X(γ(t)) = t γ(t) for a smooth curve γ. In terms of local coorinates this equation is written as X k (x(t)) = t x k(t), k = 1, 2,..., n. By the theory of orinary ifferential equations this system has locally, at a neighborhoo of an initial point p = γ(0), a unique solution. However, in general the solution oes not nee to exten to < t < + except in the case when M is a compact manifol. The (local) solution γ is calle an integral curve of X through the point p. The integral curves for a vector fiel X efine a (local) flow on the manifol M. This is a (local) map f : R M M given by f(t, p) = γ(t) where γ is the integral curve through p. We have the ientity (1) f(t + s, p) = f(t, f(s, p)), which follows from the uniqueness of the local solution to the first orer orinary ifferential equation. In coorinates, an t f k(t, f(s, x)) = X k (f(t, f(s, x))) t f k(t + s, x) = X k (f(t + s, x)). Thus both sies of (1) obey the same ifferential equation. Since the initial conitions are the same, at t = 0 both sies are equal to f(0, f(s, x)) = f(s, x), the solutions must agree. Denoting f t (p) = f(t, p), observe that the map R Diff loc (M), t f t, is a homomorphism, f t f s = f t+s.

9 Thus we have a one parameter group of (local) transformations f t on M. In the case when M is compact we actually have globally globally efine transformations on M. Example Let X(r, φ) = ( r sin φ, r cos φ) be a vector fiel on M = R 2. The integral curves are solutions of the equations 9 x (t) = r(t) sin φ(t) y (t) = r(t) cos φ(t) an the solutions are easily seen to be given by (x(t), y(t)) = (r 0 cos(φ+φ 0 ), r 0 sin(φ+ φ 0 )), where the initial conition is specifie by the constants φ 0, r 0. The one parameter group of tranformations generate by the vector fiel X is then the group of rotations in the plane. Further reaing: M. Nakahara: Geometry, Topology an Physics, Institute of Physics Publ. (1990), sections S. S. Chern, W.H. Chen, K.S. Lam: Lectures on Differential Geometry, Worl Scientific Publ. (1999), Chapter 1.