Notes for Math 535 Differential Geometry Spring Francis Bonahon. Department of Mathematics, University of Southern California

Notes for Math 535 Differential Geometry Spring 2016 Francis Bonahon Department of Mathematics, University of Southern California

Date of this version: April 27, 2016 c Francis Bonahon 2016

CHAPTER 1 A crash course in pointset topology Topological spaces. Definition 1.1. A topological space consists of a set X and of a family T of subsets of X such that: (1) the empty set and the whole space X are both elements of T ; (2) if {U i } i I is a (possibly infinite) family of elements of T, their union i I U i is also in T. (3) if {U i } i=1,2,...,n is a finite family of elements of T, their intersection i=1,2,...,n U i is also in T. The family T is then called a topology on the set X. The elements of T are called open subsets of X for this topology. Recall that, given a family {X i } i I of sets X i indexed by a set I, their union i I is the set of all x for which there exists an i I with x X i. Their intersection i I is the set of all x such that x X i for every i I. Note that I does not need to be finite. Example 1.2 (Fundamental example). Let (X, d) be a metric space, namely a set X endowed with a function d: X X R such that (1) d(x, y) 0 for every x, y; (2) d(x, y) = 0 if and only if x = y; (3) d(y, x) = d(x, y) for every x, y; (4) d(x, z) d(x, y) + d(y, z) for every x, y, z. Given x in such a metric space X and given ε > 0, let the open ball of radius ε centered at x be B(x, ε) = {y X; d(x, y) < ε}. Define T by the property that U X is an element of T if and only if, for every x U, there exists a ball B(x, ε) which is contained in U. Then, T is a topology for X. Definition 1.3. If x is an element of a topological space X, a neighborhood of x is a subset W X for which there exists an open subset U of X with x U W. Note that the terminology is not standard. This one is used in most research articles. Many textbook require a neighborhood to be open, namely just an open subset containing x. Definition 1.4. A topological space X is Hausdorff if, for every x, y X with x y, there exists a neighborhood U of x and a neighborhood V of y such that U V =. For instance, the topology of a metric space is always Hausdorff. 1

2 1. A CRASH COURSE IN POINTSET TOPOLOGY Definition 1.5. A subset A of a topological space X is dense if every nonempty open subset of X contains an element of A. The topological space X is separable if it contains a dense subset which is countable. For instance, R n with the topology associated to the metric d(x, y) = x y is separable, since it contains the dense set Q n. Continuous functions. Definition 1.6. Let f: X Y be a map between two topological spaces. The map f is continuous if, for every open subset U of Y, its pre-image f 1 (U) is an open subset of X. Recall that the pre-image of U is f 1 (U) = {x X; f(x) U}. For metric spaces, this definition of continuity turns out to be equivalent to a more familiar one: Proposition 1.7 (ε δ definition of continuity). Let f : X Y be a map between metric spaces (X, d X ) and (Y, d Y ). Then f is continuous in the above sense if and only if, for every x X and every ε > 0, there is a δ > 0 such that d Y (f(x), f(x )) < ε for every x X with d X (x, x ) < δ. Definition 1.8. A homeomorphism between two topological spaces X and Y is a bijection f: X Y such that f and f 1 are both continuous. If f : X Y is a homeomorphism and U is a subset of X, it follows from definitions that U is open in X if and only if f(u) is open in Y. Therefore, f establishes a dictionary, which enables us to translate any topological property of X to a similar property of Y. A homeomorphism is therefore an isomorphism of topological spaces. The subspace topology. Let X be a topological space, with topology T. If X is a subset of X, it is immediate that is a topology for X. T = {U X ; U T, U = X U} Definition 1.9. The topology T is the subspace topology induced on X by the topology of X. The quotient topology. Let p: X Y be a surjective map from a topological space X to a set Y. This situation typically arises when (and exactly when) Y is the quotient of X under an equivalence relation, namely when Y is the set of equivalence classes of, and when p is the natural projection. The topology of X induces a topology T on Y, defined by T = {U Y ; p 1 (U) open in X}. Definition 1.10. The topology T is the quotient topology induced on Y by the topology of X. The quotient topology is specially designed for the following property. Proposition 1.11. Let f: X Z be a map inducing a map g: Y Z in the sense that f = g p. If Z is a topological space and if f is continuous, then g is continuous for the quotient topology on Y.

CHAPTER 2 Review of differentiable functions in R n The real n space R n is the set of all n uples (x 1, x 2,..., x n ) where each x i is a real number. The norm of a vector v = (v 1, v 2,...,v n ) R n is v = v1 2 + v2 2 + + v2 n Definition 2.1. The open ball of radius ε centered at x R n is B(x, ε) = {y R n ; x y < ε} Definition 2.2. A subset U R n is open if, for every x U, there is an ε > 0 such that B(x, ε) U. Now, consider U open in R n, and a function f: U R p (x 1,..., x n ) (f 1 (x 1,..., x n ),..., f p (x 1,...,x n )) Definition 2.3. The function f: U R p is (infinitely) differentiable at x U if all the partial derivatives x i1 x i2... x ik f j (x) exist. A differentiable function f: U R p is one that is differentiable at each x U. Note that one needs U to be open for the partial derivatives to make sense. 1. The tangent map Definition 2.4. The derivative, or differential map, or tangent map (preferred terminology) of f: U R p at x U is the linear map T x f: R n R p whose matrix is ( ) fj (x) = x i f 1 f x 1 (x) 1 f x 2 (x)... 1 x n (x) f 2 x 1 (x)..................... f p f x 1 (x) p f x 2 (x)... p x n (x) Proposition 2.5 (Linear Approximation). The tangent map T x f is the only linear map L: R n R p such that f(x + h) f(x) L(h) lim = 0 h 0 h 3

4 2. REVIEW OF DIFFERENTIABLE FUNCTIONS IN R n Geometric interpretation of the tangent map. Given x and v R n, there is always a small parametrized curve α: ( ε, ε) U, namely a differentiable map defined over the open interval ( ε, ε) R, such that α(0) = x and α (0) = v. For instance, one can take α(t) = x + tv. If, in addition, we have a function f: U R p, the parametrized curve α in U gives a parametrized curve f α: ( ε, ε) R p in R p, defined by f α(t) = f(α(t)). Note that f α(0) = f(x). Let us consider its tangent vector. Proposition 2.6. T x f(v) = (f α) (0). This provides an interpretation of the tangent map T x f : R n R p in terms of parametrized curves: To determine T x f(v), pick any parametrized curve α in U with α(0) = x and α (0) = v and consider the parametrized curve f α in R p ; then T x f(v) is the tangent vector (f α) (0) to this curve. An immediate application is the following. Let U be open in R m, let V be open in R n, let W be open in R p and consider two functions f : U V and g: V W. For x U, we have tangent maps T x f: R m R n, T f(x) g: R n R p and T x (g f): R m R p. Proposition 2.7 (Generalized Chain Rule). T x (g f) = T f(x) g T x f. 2. Diffeomorphims and the Inverse Function Theorem Definition 2.8. A diffeomorphism between two open subsets U and V of R n is a bijection f: U V such that both f and f 1 are differentiable. Proposition 2.9. If f : U V is a diffeomorphism, then the tangent map T x f: R n R n is a linear isomorphism. This result has a converse statement, which will play a very important role in the course. Theorem 2.10 (Inverse Function Theorem). Let f: U V be a differentiable map between two open subsets U, V of R n. Suppose that, at the point x U, the tangent map T x f: R n R n is an isomorphism. Then there is an open neighborhood U U of x such that f(u ) is open in R n such that the restriction f U : U f(u ) is a diffeomorphism. The proof can be found in any good textbook in (second semester) real analysis or in advanced calculus. We will assume this result in the class.

CHAPTER 3 Submanifolds of R n 1. Definitions Definition 3.1. An m dimensional submanifold of R n is a subset M R n such that, for every x M, there exists an open neighborhood U of x in R n and a diffeomorphism ϕ: U V from U to an open subset V R n = R m R n m such that ϕ(u M) = V (R m {0}) x. Recall that a neighborhood of x is a subset containing an open subset containing Example 3.2. An n dimensional submanifold of R n is exactly an open subset of R n. Example 3.3. A 0 dimensional submanifold of R n is a discrete subset, namely such that every point x M has a neighborhood U such that U M = {x}. (Recall that R 0 is a vector space of dimension 0, namely R 0 = {0}.) Example 3.4. Let f : U R p be a differentiable function defined on an open subset U of R n. Then the graph G f = {(x, y) U R p ; y = f(x)} is an n dimensional submanifold M of R n+p. We will see in 4 that every submanifold of R n+p is locally of this type, in a sense which will be precised there. Example 3.5. Let f : R 2 R be defined by f(x, y) = x 2 + y 2. The graph G f is not a submanifold of R 3. (Note that f is not differentiable at (0, 0).) Example 3.6. The n dimensional sphere { n } S n = (x 0, x 1,..., x n ) R n+1 ; x 2 i = 1 is an n dimensional submanifold of R n+1. 2. Tangent vectors Definition 3.7. Let M be an m dimensional submanifold of R n. A vector v R n is tangent to M at x if there exists a curve α: ( ε, ε) R n with image contained in M such that α(0) = x and α (0) = v. The set T x M: R n R p of tangent vectors of M at x is the tangent space of M at x. Proposition 3.8. The tangent space T x M is an m dimensional linear subspace of R m. 5 i=0

6 3. SUBMANIFOLDS OF R n 3. Submanifolds defined by equations Theorem 3.9. Let f: U R p be a function defined on an open subset U R n. For y 0 R p, consider M = {x U; f(x) = y 0 }. Suppose that, for every x M, the tangent map T x f: R n R p is surjective. Then M is a submanifold of R n of dimension n p. In other words, if M U open R n is defined by p equations, and it these equations are well-behaved, then M is a submanifold of dimension n p. Proposition 3.10. For data as above, the tangent space T x M is the kernel of the tangent map T x f: R n R p. Example 3.11. The n dimensional sphere { n } S n = (x 0, x 1,..., x n ) R n+1 ; x 2 i = 1 is an n dimensional submanifold of R n+1. Example 3.12. Let M n (R) = R n2 be the set of n by n matrices. The subsets GL n (R) = {A M n (R); deta 0} SL n (R) = {A M n (R); deta = 1} O n = {A M n (R); AA t = Id} SO n = {A M n (R); AA t = Id and deta = 1} = {A O n ; deta > 0} are submanifolds of M n (R) = R n2, of respective dimensions n 2, n 2 1, 1 2n(n 1) and 1 2n(n 1). i=0 4. Submanifolds as graphs An example of submanifold of R n was the graph G f = {(x, y) U R n m ; y = f(x)} of a function f: U R n m, where U is an open subset of R m. We show that this example is actually quite general, because every submanifold of R n is locally of this type. Theorem 3.13. Let M be an m dimensional submanifold of R n. For every x M, after possible reshuffling of the coordinates of R n, there exists a neighborhood U of x in R n and a differentiable function f: V R n m, defined on an open subset V of R m, such that U M is equal to the graph G f of the function f. 5. Coordinate charts and atlases The following is a preparation for the definition of abstract manifolds in the next chapter. Proposition 3.14. Let M be an m dimensional submanifold of R n. Then there exists a family A = {(U i, ϕ i ); i I} where: (1) U i is an open subset of M (for the subspace topology) for every i I, and M = i I U i; (2) for every i I, ϕ i : U i V i is a homeomorphism from U i to an open subset V i of R m ;

5. COORDINATE CHARTS AND ATLASES 7 (3) for every i, j I, the composition ϕ j ϕ 1 i ϕ i(u i U j) : ϕ i (U i U j ) ϕ j (U i U j ) is a diffeomorphism. For the last condition, note that ϕ i (U i U j ) is open in V i for the subspace topology, and therefore is open in R m. The same holds for ϕ j (U i U j ), so that the requirement that the restriction ϕ j ϕ 1 i ϕ i(u i U j) be a diffeomorphism really makes sense. The pairs (U i, ϕ i ) are coordinate charts for M, and the collection A of charts covering M is an atlas.

CHAPTER 4 Differentiable manifolds 1. Manifolds Definition 4.1. A topological manifold of dimension n is a separable Hausdorff topological space M such that, for every x M, there is an open neighborhood U of x and a homeomorphism ϕ: U V from U to an open subset V of R n. Definition 4.2. A differentiable atlas for the topological manifold M is a family A = {(U i, ϕ i ); i I} where: (1) U i is an open subset of M for every i I, and M = i I U i; (2) for every i I, ϕ i : U i V i is a homeomorphism from U i to an open subset V i of R m ; (3) for every i, j I, the composition ϕ j ϕ 1 i ϕ i(u i U j) : ϕ i (U i U j ) ϕ j (U i U j ) is a diffeomorphism. The (U i, ϕ i ) are the charts, or coordinate patches of the atlas A, and the diffeomorphisms ϕ j ϕ 1 i ϕ i(u i U j) are its changes of charts or coordinate changes. Very often, we will just write ϕ j ϕ 1 i only defined on ϕ i (U i U j ). for ϕ j ϕ 1 i ϕ i(u i U j), since this map is Definition 4.3. A differentiable structure on the topological manifold M is a differentiable atlas A which is maximal, namely for which there is no other differentiable atlas A A with A A. A differentiable manifold of dimension n is an n dimensional topological manifold M endowed with a differentiable structure A. To explain the use of maximal atlases, consider the following easy lemma. Lemma 4.4. Any differentiable atlas A is contained in a unique maximal atlas, which is equal to the set A max of all pairs {(U, ϕ ) where U is open in M, ϕ : U V is a homeomorphism from U to an open subset V of R n, and where ϕ ϕ 1 is differentiable for every (U, ϕ) A. An immediate consequence of Lemma 4.4 is that every (not necessarily maximal) atlas on the topological manifold M defines a unique differentiable structure for M. This is more useful in practice, since a maximal atlas is really huge whereas a manifold usually admits atlases of a more reasonable size. A more natural approach to defining differentiable structures would have been to say that two atlases A and A are compatible if their union A A is also a differentiable atlas, namely if any change of chart ϕ ϕ 1 between a chart (U, ϕ) A and a chart (U, ϕ ) A is a diffeomorphism. We could then define a differentiable structure as an equivalence class of the relation is compatible with over the set of all possible atlases for M. Lemma 4.4 shows that this definition is equivalent to the first one. Maximal atlases were actually introduced as an elegant but impractical way of avoiding the use of this compatibility condition. 9

10 4. DIFFERENTIABLE MANIFOLDS 2. Some examples Example 4.5. Every open subset U of R n is a differentiable manifold, with atlas A = {(U, Id U )}, where Id U : U U is the identity map. Example 4.6. More generally, every submanifold of R n admits a natural differentiable structure by (the proof of) Proposition 3.14. Example 4.7. Consider the real projective n space RP n, defined as RP n = {lines through the origin in R n+1 } = (R n+1 {0})/ where the equivalence relation on R n+1 {0} is defined by the property that x y λ R, y = λx. The set RP n = (R n+1 {0})/ is endowed with the quotient topology. For i = 0, 1,..., n, let U i denote the image in RP n of {(x 0, x 1,...,x n ) R n+1 ; x i 0} R n+1 {0}. Let ϕ i : U i R n associate to the class of (x 0, x 1,...,x n ) the element ( x0 x i, x1 x i,..., xi 1 x i, xi+1 x i,..., xn x i ) R n. Then, A = {(U 0, ϕ 0 ), (U 1, ϕ 1 ),..., (U n, ϕ n )} is a differentiable atlas for RP n. This turns RP n into a differentiable manifold of dimension n. Example 4.8. The complex projective plane CP n is the space of complex lines through the origin in C n+1. Namely, CP n is equal to the quotient space (C n+1 {0})/ where x y λ C, y = λx, and is endowed with the quotient topology. The same formulas as for the real projective space provide charts ϕ i : U i C n = R 2n, which endows CP n with the structure of a differentiable manifold of dimension 2n. Example 4.9 (Products of manifolds). If {(U i, ϕ i } i I is a differentiable atlas for M and {(V j, ψ j } j J is a differentiable atlas for N, then {(U i V j, ϕ i ψ j } (i,j) I J is a differentiable atlas for the product M N (with the product topology). In particular, if M and N are differentiable manifolds of dimensions m and n, respectively, then M N is a differentiable manifold of dimension m + n. 3. Differentiable functions between manifolds Definition 4.10. Let M be a differentiable manifold of dimension m with atlas {(U i, ϕ i )} i I, and let N be a differentiable manifold of dimension n with atlas {(V j, ψ j )} j J. A function f : M N is differentiable at x if f is continuous and if, for any i I and j J with x U i and f(x) V j, the map ψ j f ϕ 1 i : ϕ i (U i f 1 (V j ) ψ j (V j ) is differentiable at x in the sense of Chapter 2, which makes sense since it goes from an open subset of R m to an open subset of R n. The map f: M N is differentiable if it is differentiable at every x M. Lemma 4.11. This does not depend on the choice of i, j such that x U i and f(x) V j. To a large extent, the notion of differentiable atlas was introduced precisely to make sense of differentiable maps f: M N. Definition 4.12. A diffeomorphism from the manifold M to the manifold N is a bijection f: M N such that both f and f 1 are differentiable. The manifolds M and N are diffeomorphic if there exists a diffeomorphism f: M N.

4. CONSTRUCTING MANIFOLDS AS QUOTIENT SPACES 11 Example 4.13. Let M be a differentiable manifold of dimension m with atlas {(U i, ϕ i )} i I. Then, each chart ϕ i : U i ϕ i (U i ) is a diffeomorphism between the manifold U i, endowed with the single chart differentiable atlas {(U i, ϕ i )} and the open subset ϕ i (U i ) of R m. Diffeomorphic manifolds are undistinguishable for any property involving differentiable maps. This is a stronger property than being homeomorphic, which involves only continuity properties. For instance Milnor showed around 1960 that there are exactly 28 differentiable manifolds that are homeomorphic to the 7 dimensional sphere S 7 and not diffeomorphic to each other. 4. Constructing manifolds as quotient spaces Definition 4.14. An action of the group Γ on the manifold M is a group homomorphism ρ from Γ to the group Diff(M) of diffeomorphisms of M. In general, if γ Γ and x M, we write γx = ρ(γ)(x) for the image of x under the diffeomorphism ρ(γ). Definition 4.15. The quotient space M/Γ of the action of Γ on the manifold M is the set of equivalence classes of the equivalence relation defined by x y γ Γ, y = γx, endowed with the quotient topology. For an arbitrary action, the quotient space M/Γ may not even be Hausdorff. Definition 4.16. The action of Γ on M is discontinuous if, for every compact subset K of M, the set {γ Γ; K γk } is finite. Definition 4.17. The action of Γ on M is free if γx x for every x M and every γ Γ {Id}. Lemma 4.18. Let X be a Hausdorff locally compact topological space, and let the group Γ act freely and discontinuously on X. Then, every x X admits a neighborhood U such that the restriction p U : U p(u) of the canonical projection p: X X/Γ is a homeomorphism. In addition, X/Γ is Hausdorff. Theorem 4.19. Suppose that the action of Γ on he manifold M is free and discontinuous. Then the quotient space M/Γ is a topological manifold. In addition, the maximal atlas of M contains an atlas A = {(U i, ϕ i )} i I such that, for every i I, the canonical projection p: M M/Γ restricts to a homeomorphism p Ui : U i p(u i ) in such a way that A = {(p(u i ), ϕ i (p Ui ) 1 )} i I is a differentiable atlas for M/Γ. In particular, if Γ acts freely and discontinuously on the manifold M, then M/Γ inherits a natural differentiable structure. Example 4.20. Let Z n act on M by (k 1, k 2,..., k n )(x 1, x 2,... x n ) = (x 1 + k 1, x 2 + k 2,...,x n + k n ) for every (k 1, k 2,..., k n ) Z n and (x 1, x 2,... x n ) R n. The quotient manifold R n /Z n is the n dimensional torus, and is diffeomorphic to the product S 1 S 1 S 1 of n copies of the circle S 1. Example 4.21. Let Z act on the plane R 2 by k(x, y) = (x+k, y). The quotient manifold R 2 /Z is the cylinder, diffeomorphic to S 1 R. Example 4.22. Let Z act on the plane R 2 by k(x, y) = (x + k, ( 1) k y). The quotient manifold R 2 /Z is the (open) Möbius strip.

12 4. DIFFERENTIABLE MANIFOLDS Example 4.23. Let Z 2 act on the plane R 2 by (k 1, k 2 )(x 1, x 2 ) = (x 1 +k 1, ( 1) k1 x 2 + k 2 ). The quotient manifold R 2 /Z 2 is the Klein bottle. Example 4.24. Let Z 2 act on the n dimensional sphere S n R n+1 by sending its generator to the diffeomorphism x x. The quotient manifold S n /Z 2 is diffeomorphic to the projective space RP n.

CHAPTER 5 Tangent vectors 1. Tangent vectors for manifolds We now extend the notion of tangent vector to the abstract framework of manifolds. 1.1. Tangent vectors as equivalence class of curves. Definition 5.1. A parametrized curve in the manifold M is a differentiable map α: I M where I is an open interval in R. Let M be a differentiable manifold of dimension n, with atlas A = {(U i, ϕ); i I}, and let x M. Consider the set C x of all parametrized curves α: I M such that 0 I and α(0) = x. Choose a chart (U i, ϕ i ) of A such that x U i. Because U i is open, a parametrized curve α C x defines a parametrized curve ϕ i α in R n defined on a small neighborhood of 0. Define an equivalence relation on C x by the property that α β exactly when (ϕ i α) (0) = (ϕ i β) (0). Lemma 5.2. The equivalence relation on C x is independent of the choice of the chart (U i, ϕ i ) containing x. Definition 5.3. The tangent space T x M of the manifold M at x is the set of equivalence classes of the relation on C x. An element of T x M is a tangent vector of M at x. Lemma 5.4. If the chart (U i, ϕ i ) contains x, the map Φ i : T x M R n defined by α (ϕ i α) (0) is bijective. In addition, for any other chart (U j, ϕ j ) containing x, the map Φ j Φ 1 i : R n R n coincides with the tangent map T ϕi(x)ϕ j ϕ 1 i of the coordinate change diffeomorphism ϕ j ϕ 1, and in particular is linear. Corollary 5.5. The tangent space T x M has a unique structure as a vector space such that, for every chart (U i, ϕ i ) containing x, the above map Φ i : T x M R n is a linear isomorphism. Corollary 5.6. If M is a submanifold of R n, there is a natural one-to-one correspondence between the tangent space T x M defined here and the tangent space T x M R n defined earlier. In practice, we will say that the tangent vector v T x M represented by the curve α: I M with α(0) = x is the vector tangent to α at t = 0 (or at x), and we will write that v = α (0). 13

14 5. TANGENT VECTORS 1.2. Tangent vectors as derivations. Definition 5.7. Let C(M) denote the vector space of all differentiable functions f: M R. A derivation at the point x M is a linear map D: C(M) R that satisfies the following product rule: for every f, g C(M). D(fg) = D(f)g(x) + f(x)d(g) A typical example is provided by the directional derivative D v along a tangent vector v T x M, defined by D v (f) = (f α) (0) for an arbitrary curve α: I M such that α(0) = x and α (0) = v. Theorem 5.8. The map v D v defines a linear isomorphism between the tangent space T x M and the vector space of all derivations at x. Example 5.9. If U is open in R n, the derivation x i corresponds to the basis vector (0,...,0, 1, 0..., 0) where the 1 is in the i th position. 2. The tangent map to a map between manifolds Proposition 5.10. Let f: M N be a differentiable map between manifolds. For every x M, there is a unique linear map T x f: T x M T f(x) N such that, if v T x M is represented by the parametrized curve α: ( ε, ε) M with α(0) = x, then T x f(v) T f(x) N is represented by the curve f α: ( ε, ε) N. Proposition 5.11 (Chain rule). Given three manifolds M, N, P and two differentiable maps f: M N and g: N P, then T x (g f) = T f(x) g T x f. Corollary 5.12. If f : M N is a diffeomorphism between manifolds, the tangent map T x f: T x M T f(x) N is a linear isomorphism. As in the case of open subsets of R n, this fact has a local converse. Theorem 5.13 (Inverse Function Theorem). Let f: M N be a differentiable map between manifolds M and N of the same dimension. If the tangent map T x f: T x M T f(x) N is an isomorphism, then there exists an open neighborhood U of x and an open neighborhood V of f(x) in N such that f restricts to a diffeomorphism U V. 3. Submanifolds of manifolds Definition 5.14. Let M be a manifold of dimension m. A submanifold of M of dimension n is a subset N M such that, for every x N, there exists an open neighborhood U of x in M and a diffeomorphism ϕ: U V from U to an open subset V of R n such that in R m = R n R m n. ϕ(u M) = V (R n {0}) As in the case of submanifolds of R m, the differentiable structure of M induces a differentiable structure on N.

4. THE TANGENT BUNDLE 15 Proposition 5.15. If N is a submanifold of the manifold M, the inclusion map i: N M defined by i(x) = x is differentiable. Complement 5.16. The tangent map T x i: T x N T x M of the inclusion map i: N M is injective, and identifies T x N to a linear subspace of T x M. 3.1. Immersions and embeddings. Definition 5.17. A differentiable map f: M N between two manifolds is an immersion if the tangent map T x f: T x M T f(x) N is injective at each x M. Definition 5.18. An embedding of the manifold M into the manifold N is an immersion f: M N for which the range restriction M f(n) is a homeomorphism. Theorem 5.19. Let f: M N be an embedding between differentiable manifolds. Then the image f(m) is a submanifold of N, and the restriction M f(m) of f is a diffeomorphism. Complement 5.20. If x M and y = f(x), the tangent space T y f(m) T y N of the submanifold f(m) N is the image of T x M under the tangent map T x f 3.2. Critical points, regular values. Definition 5.21. A critical point of the differentiable map f : M N is a point x M such that the tangent map T x f: T x M T f(x) N is not surjective. A regular point is a point x M that is not critical. Definition 5.22. A critical value of the differentiable map f : M N is a point y N that is the image of a critical point. A regular value is a point y N that is not critical. In other words, y N is regular if every x f 1 (y) is a regular point. Remark 5.23. If the dimension of M is smaller than the dimension of N, the regular values of f: M N are just the points of N which are not in the image f(m). Theorem 5.24. Let f: M N be a differentiable map from a manifold M of dimension m to a manifold N of dimension n. If y is a regular value of f, then its preimage f 1 (y) is a submanifold of M of dimension m n (possibly empty if y f(m)). Complement 5.25. If y is a regular value of f: M N, and if P = f 1 (y), the tangent space T x P T x M of the submanifold P M is equal to the kernel of the tangent map T x f: T x M T y N. set 4. The tangent bundle Definition 5.26. The tangent bundle of the m dimensional manifold M is the TM = {(x, v); x M, v T x M}. The tangent bundle comes endowed with a natural projection p: TM M, defined by p(x, v) = x. Let {(U i, ϕ i )} i I be an atlas for M. Set Ũi = p 1 (U i ) TM, and define ϕ i : Ũi V i R m R 2m by the property that ϕ i (x, v) = (ϕ(x), T x ϕ(v)).

16 5. TANGENT VECTORS We endow TM with the topology for which U TM is open if and only if ϕ i (U U i ) is open in R m for every i I. Lemma 5.27. The family {(Ũi, ϕ i )} i I is a differentiable atlas for TM. As a consequence, the tangent bundle TM is a differentiable manifold of dimension 2m. 5. Vector fields Definition 5.28. A vector field on the manifold M is a differentiable map X: M TM such that p X = Id M. In other words, a vector field associates a tangent vector X x T x M to each x M, in such a way that the map x X x is differentiable. The tangent bundle non-sense is here to make sense of the differentiability property. 5.1. Integrating vector fields. Definition 5.29. A flow on the manifold M is a differentiable map ϕ: M R M such that ϕ(x, t + t ) = ϕ ( ϕ(x, t), t ) for every x M and t, t R. This is better expressed in terms of the map ϕ t : M M defined by ϕ t (x) = ϕ(x, t); then the above property is equivalent to the equality ϕ t+t = ϕ t ϕ t. In particular, each ϕ t is a diffeomorphism, and ϕ 0 = Id M and ϕ 1 t = ϕ t. Sometimes, a flow is also called a 1 parameter group of diffeomorphisms (ϕ t ) t R of M. A flow defines a vector field X by the property that X x = d dt ϕ t(x) t=0 for every x X. Proposition 5.30. Let M be a compact manifold, and let X be a vector field on M. Then, there exists a unique flow (ϕ t ) t R such that X x = d dt ϕ t(x) t=0 for every x X. There exists local version of this result when M is not compact. 5.2. The Lie bracket of two vector fields. Let X and Y be two vector fields. Given a function f C(M) and a point x M, we can interpret the vector Y x T x M defined by Y as a derivation, and consider the directional derivative Y x (f) R. The map x Y x (f) is now a new function Y (f) of x, and we can consider its directional derivative X x (Y (f)) = (XY ) x (f). In general, the map f (XY ) x (f) is not a derivation, because it does not satisfy the product rule. Lemma 5.31. Let X and Y be two vector fields on M. Then, there exists another vector field [X, Y ] such that [X, Y ] x (f) = X x (Y (f)) Y x (X(f)) for every function f: M R and every point x M. The vector field [X, Y ] is the Lie bracket of X and Y. Proposition 5.32. The Lie bracket satisfies the following properties: (1) [X, Y ] is a linear function of X and Y, in the sense that the maps X [X, Y ] and Y [X, Y ] are linear; (2) [Y, X] = [X, Y ]; (3) [X, fy ] = X(f)Y + f[x, Y ] for every function f; (4) (Jacobi identity) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0.

5. VECTOR FIELDS 17 5.3. The Frobenius theorem. Theorem 5.33. Let X 1, X 2,..., X p be p vector fields in the m dimensional manifold M, such that at every x M the vectors X 1x, X 2x,..., X px are linearly independent in the tangent space T x M. (In particular, p m.) Let V x T x M denote the p dimensional linear subspace generated by X 1x, X 2x,..., X px. The following two properties are equivalent: (i) Every x M is contained in a p dimensional submanifold N M such that T y N = V y for every y N; (ii) For every i, j and every x M, the Lie bracket [X i, X j ] x T x M is contained in the subspace V x generated by the vectors X kx. Example 5.34. In R 3, there is no 2 dimensional submanifold N that is tangent to the vectors X 1 = x and X 2 = y + x z at every (x, y, z) N.

CHAPTER 6 Differentiable partitions of unity 1. Partitions of unity Definition 6.1. Let {U i ; i I} be an open covering of the manifold M. A (differentiable) partition of unity subordinate to this open covering is a family of differentiable functions ψ j : M R, defined for j ranging over an index set J, such that: (1) ψ j (x) for every j J and x X; (2) for every j J the support Supp(ψ j ) of ψ j, namely the closure of {x M; ψ j (x) 0}, is compact and contained in some U i ; (3) for every compact subset K of M, the set {j J; K Supp(ψ j ) } is finite; (4) for every x M, j J ψ j(x) = 1. Note that the sum in Condition 4 is finite by Condition 3. Theorem 6.2. Let {U i ; i I} be an open covering of the manifold M. Then, there exists a partition of unity {ψ j : M R; j J} subordinate to {U i ; i I}. A key ingredient in the proof, using the fact that M is separable, is the following. Lemma 6.3. Let M be a locally compact, separable topological space. Then there is a countable family of subsets {K n ; n N} such that: (1) for every n N, K n is compact and contained in the interior of K n+1 ; (2) n N K n = M. Such a family {K n ; n N} is called an exhaustion of M by compact subsets. Partitions of unity are a very useful technical tool to construct functions on manifolds. The following section illustrates an application. 2. Manifolds as submanifolds of R n Theorem 6.4. If M is a compact manifold of dimension m, there exists a number n and an embedding f: M R n. The proof goes as follows. Let {(U i, ϕ i )} i I be an atlas for M, and let ψ j : M R} j J be a partition of unity subordinate to the covering by the U i. Since M is compact and since the collection of the supports of the ψ j is locally finite, the index set J is finite. Assume that J = {1, 2,..., p} with loss of generality. By definition of partitions of unity, for every j J, the support of ψ j is contained in some U ij. Then we can take n = p + pm and define f by f(x) = ( ψ 1 (x), ψ 2 (x),..., ψ p (x), ψ 1 (x)ϕ i1 (x), ψ 2 (x)ϕ i2 (x),..., ψ p (x)ϕ ip (x) ) 19

20 6. DIFFERENTIABLE PARTITIONS OF UNITY with image in R n = R R R R m R m R m. The result is actually true even if M is not compact, but more unpleasant to prove. Corollary 6.5. Every compact manifold is diffeomorphic to a submanifold of some R n.

CHAPTER 7 Sard s theorem and applications 1. Sard s Theorem Definition 7.1. Let M be an m dimensional manifold with atlas {(U i, ϕ i )} i I. A subset A M has measure 0 in M if, for every i I, ϕ i (A U i ) has Lebesgue measure 0 in R m. A subset B M has full measure if A = M B has measure 0 in M. Because every differentiable map from R m to R m sends a set of measure 0 to a subset of measure 0, this definition is independent of the atlas used. Theorem 7.2 (Sard s theorem). Let f: M N be a differentiable map between differentiable manifolds. Then the set of regular values of f has full measure in N. Corollary 7.3. Let f: M N be a differentiable map between differentiable manifolds, such that the dimension of M is strictly less than the dimension of M. Then the image f(m) has measure 0 in N. 2. Application: Embedding manifolds in small R n Theorem 7.4. Let M be a compact manifold of dimension m. Then there exists an embedding of M in R 2m+1. The proof is by downward induction, starting from Theorem 6.4. Suppose that we have an embedding f : M R n with n > 2n + 1. For every v S n 1, let L v = R n 1 be the hyperplane orthogonal to v in R n, and let π v : R n L v be the orthogonal projection to L v. Lemma 7.5. If n > 2m+1, then the set of those v for which π v f: M L v = R n 1 is an embedding has full measure in S n 1. Lemma 7.5 immediately proves Theorem 7.4 by induction. Lemma 7.5 follows from an application of Sard s Theorem to two functions F : M M S n 1 and G: TM S n 1 defined as follows. The function F: M M S n 1 is defined outside of the diagonal = {(x, y) M M; x = y} by F(x, y) = f(x) f(y) f(x) f(y). The function G: TM S n 1 is defined on the open subset TM = {(x, u); x M, u T x M, u 0} of the unit tangent bundle TM by G(x, u) = T xf(u) T x f(u). Note that one needs f to be an embedding for F and G to be defined. By construction, π v f: M L v is injective exactly when v S n 1 is not in the image of F, 21

22 7. SARD S THEOREM AND APPLICATIONS and it is an immersion exactly when v is not in the image of G. Since the domains of F and G are manifolds of dimension 2m < n 1, Lemma 7.5 immediately follows from these observations and from Corollary 7.3. A slight variation of these arguments (using the restriction of G to the (2m 1) manifold T 1 M = {(x, u); x M, u T x M, T x f(u) = 1}) gives: Theorem 7.6. If M is a compact manifold of dimension m, there exists an immersion g: M R 2m 1.

CHAPTER 8 The alternating algebra This is pure linear algebra, in preparation for the introduction of differential forms. Let V be a vector space over R, usually finite-dimensional. Let V p = V V V denote the product of k copies of V. Definition 8.1. A p linear form, or a multilinear form of degree p, on V is a map ω : V p R such that the map V R defined by v i ω(v 1,..., v i 1, v i, v i+1,..., v p ) is linear, for every i = 1,...,p and every v 1,..., v i 1, v i+1,..., v p V. The p linear form ω : V p R is alternating if ω(v 1,..., v p ) = 0 whenever there exists i j such that v i = v j. The definition of alternating forms (also called antisymmetric) is made somewhat clearer by the following statement. Let S p denote that symmetric group, consisting of all bijections of {1, 2,..., p}. Recall that the signature homomorphism is the unique group homomorphism sign: S p Z 2 = {±1} sending any transposition to 1. Proposition 8.2. The p linear form ω : V p R is alternating if and only ω(v σ(1), v σ(2),..., v σ(p) ) = sign(σ)ω(v 1, v 2,..., v p ) for every v 1, v 2,..., v p V and every σ S p. Definition 8.3. Let Alt p (V ) denote the vector space of all alternating p linear forms ω: V p R In the special case where p = 1, note that Alt 1 (V ) is just the dual V of V. When p = 0, we decide that Alt 0 (V ) = R by convention. 1. The wedge product Definition 8.4. The wedge product of the forms α Alt p (V ) and β Alt q (V ) is the (p + q) form α β Alt p+q (V ) defined by α β(v 1, v 2,...,v p+q ) = 1 sign(σ)α(v σ(1), v σ(2),...,v σ(p) )β(v σ(p+1), v σ(p+2),..., v σ(p+q) ) p!q! σ S p+q for every v 1, v 2,..., v p+q V. In the above definition, the sum over all σ S p+q is designed to make sure that the right hand term is really alternating. The 1 p!q! factor is introduced to guarantee the following: 23

24 8. THE ALTERNATING ALGEBRA Proposition 8.5. If α Alt p (V ), β Alt q (V ), and γ Alt r (V ), then in Alt p+q+r (V ) α (β γ) = (α β) γ Proposition 8.6. If α Alt p (V ) and β Alt q (V ), then β α = ( 1) pq α β. All of this makes sense even when p = 0 (or q = 0, etc... ), in which case the wedge product with the constant α Alt 0 (V ) = R is just the multiplication by the corresponding constant. One often combines all Alt p (V ) into a single vector space Alt (V ) = Alt p (V ). p=0 Then, the wedge product endows Alt (V ) with the structure of an algebra, called the alternating algebra of the vector space V. If you already know what the exterior algebra Λ (V ) of a vector space V is, the alternating algebra Alt (V ) has a natural identification with the exterior algebra Λ (V ) of the dual space V. 2. A basis for Alt p (V ) We now assume that V is finite dimensional, with basis {e 1, e 2,..., e n }. Let {e 1, e 2,...,e n} be the dual basis for the dual space V = Alt 1 (V ). Then, for every i 1, i 2,..., i p, we can consider the element e i 1 e i 2 e i p Alt p (V ). Theorem 8.7. If {e 1, e 2,..., e n } is a basis for V, the elements {e i 1 e i 2 e i p ; 1 i 1 < i 2 < < i p p} form a basis for Alt p (V ). Corollary 8.8. If V is a vector space of dimension n, then Alt p (V ) has dimension ( n p). Note that this holds even when p = 0. Also, Alt p (V ) = 0 if p > n or p < 0. 3. Linear maps induced by linear maps Let f: V M be a linear map between two vector spaces V and W. It induces a linear map Alt p (f): Alt p (W) Alt p (V ) where, for every ω Alt p (W), Alt p (f)(ω) Alt p (V ) is defined by the natural formula Alt p (f)(ω)(v 1, v 2,..., v p ) = ω(f(v 1 ), f(v 2 ),..., f(v p )) for every v 1, v 2,..., v p V. The functorial notation where Alt p (f) denotes a map between the vector spaces Alt p (W) and Alt p (V ) takes a while to get used to, but eventually turns out to be very convenient when used systematically. Note that, when p = 1, Alt 1 (f): Alt 1 (W) Alt 1 (V ) is just the dual map f : W V of the linear map f: V W. When p = 0, Alt 0 (f): Alt 0 (W) Alt 0 (V ) is by convention the identity map R R.

4. CONNECTION WITH DETERMINANTS 25 Proposition 8.9. If we have a composition of linear maps U f V g W, inducing a composition Alt p (W) Altp (g) Alt p (V ) Altp (f) Alt p (U), then Alt p (g f) = Alt p (f) Alt p (g). Proposition 8.10. If f: V W, α Alt p (W) and β Alt q (W), Alt p+q (f)(α β) = Alt p (f)(α) Alt q (f)(β). 4. Connection with determinants Consider the special case where f is a linear map from the n dimensional vector space V to itself. Then f induces a linear map Alt n (f) from Alt n (V ) = R to itself, which must be the multiplication by a certain constant. Theorem 8.11. If f: V V is represented by the n n matrix F, then the induced map Alt n (f): Alt n (V ) Alt n (V ) is just the multiplication by the determinant det(f). The machinery of alternating forms therefore provides a way to define determinants in a manner which is independent of the choice of a basis in V. This property is the crucial motivation behind the definition of differential forms.

CHAPTER 9 Differential forms in R m 1. Differential forms in R m Definition 9.1. A differential form of degree p on the open subset U of R m is a differentiable map ω: U Alt p (R m ). Let Ω p (U) denote the vector space of all differential forms of degree p on U. Let e i = (0,...,0, 1, 0,..., 0) denote the standard basis element of R m, where the 1 occurs in i th position, and let e i be the corresponding element of the dual basis of (R m ). For every x M, we can therefore write ω(x) = ω i1...i p (x)e i 1... e i p 1 i 1< <i p m where the ( n p) functions ωi1...i p : U R are differentiable. Example 9.2 (Degree 0). A differential form of degree 0 is just a differentiable function ω: U Alt 0 (R m ) = R. Example 9.3 (A fundamental degree 1 example). The differential of a function f: U R is the differential form df Ω 1 (U) defined by the property that df(x) = T x f Alt 1 (R m ) for every x U. In practice, the differential of the function f: U R is equal to m f df = e x i. i i=1 In particular, if f is the function x i, namely the function that to x U associates the coordinate x i, then df = e i. For this reason, we will henceforth write that e i = dx i. The vector spaces Ω p (U) also come with a multiplication, coming from the wedge product. Definition 9.4. The wedge product or exterior product of the differential forms α Ω p (U) and β Ω q (U) is the differential form α β Ω p+q (U) defined by the property that (α β)(x) = α(x) β(x) Alt p+q (R m ). Proposition 9.5. If α Ω p (U), β Ω q (U) and γ Ω r (U), then α (β γ) = (α β) γ and β α = ( 1) pq α β. 27

28 9. DIFFERENTIAL FORMS IN R m 2. Homomorphisms induced by maps Let f : U V be a differentiable map between open subsets U R m and V R n. For every x U, the tangent map T x f : R m R n induces a map Alt p (T x f): Alt p (R n ) Alt p (R m ). We can therefore define a map Ω p (f): Ω p (V ) Ω p (U) by the formula Ω p (f)(ω)(x) = Alt p (T x f)(ω(f(x)) for every ω Ω p (V ) and every x U. By definition, Ω p (f)(ω) Ω p (U) is the pull back of ω Ω p (V ) under the map f: U V, and Ω p (f) is the linear homomorphism Ω p (V ) Ω p (U) induced by f. Proposition 9.6. If f: U V, α Ω p (V ) and β Ω p (V ), then Ω p+q (f)(α β) = Ω p (α) Ω q (β). Proposition 9.7. If we have a composition of differentiable maps U f V g W, then Ω p (g f) = Ω p (f) Ω p (g). Let us explicitly compute the homomorphism Ω p (f): Ω p (V ) Ω p (U) induced by the map f: U V with coordinate functions f j : U R, j = 1,..., n If ω Ω p (V ) is given by ω = ω j1...j p dy j1 dy jp then Ω p (f)(ω) = = = = 1 j 1< <j p n 1 j 1< <j p n 1 j 1< <j p n 1 j 1< <j p n i 1=1 1 j 1< <j p n Ω p (f) ( ω j1...j p dy j1... dy jp ) Ω 0 (f)(ω j1...j p )Ω 1 (f)(dy j1 ) Ω 1 (f)(dy jp ) ( ωj1...j p f ) ( ) m f j1 m f dx i1 jp dx ip x i1 x ip m m i p=1 i 1=1 i p=1 ( ωj1...j p f ) f j1 x i1... f j p x ip dx i1 dx ip. Note the analogy between the expression Ω 1 (f)(dy j ) = m standard calculus formalism dy j = m i=1 i=1 f j x i dx i associated to y = f(x). f j x i dx i and the The following special case, where m = n = p, will be fundamental when we integrate differential forms. Lemma 9.8. Let U and V be open in R n and consider a differentiable function f : U V with coordinate functions f j : U R. Consider the form ω Ω n (V ) defined by ω = g dy 1 dy 2 dy n for some function g: V R. Then ( ) Ω n fj (ϕ)(ω) = g f det dx 1 dx 2 dx n. x i

3. THE EXTERIOR DERIVATIVE 29 3. The exterior derivative Definition 9.9. The exterior derivative is the linear map d: Ω p (U) Ω p+1 (U) defined by the property that, if ω = ω i1i 2...i p dx i1 dx i2 dx ip i 1<i 2< <i p then dω = i 1<i 2< <i p ( m i=1 ω i1i 2...i p dx i x i ) dx i1 dx i2 dx ip Proposition 9.10. The exterior derivative has the following properties. (1) d is linear; (2) if α Ω p (U) and β Ω q (U), d(α β) = (dα) β + ( 1) p α dβ; (3) if f: M N is differentiable, d Ω p (f) = Ω p+1 (f) d; (4) d d = 0; (5) if f Ω 0 (U) is interpreted as a function f : U R, then df Ω 1 (U) is the differential of f defined by the property that df(x) = T x f at each x U.

CHAPTER 10 Differential forms on manifolds 1. Differential forms in manifolds We now consider the case where M is a differentiable manifold of dimension m, with atlas {(U i, ϕ i )} i I. At every x U i, corresponding to y = ϕ i (x) ϕ i (U i ) R m, the tangent map T y ϕ 1 i : R m T x M induces an isomorphism Alt p (T y ϕ 1 i ): Alt p (T x M) Alt p (R m ). Definition 10.1. A differential form of degree p on the manifold M is the assignment of an element ω(x) Alt p (T x M) to each x M, in such a way that ω(x) depends differentiably on x in the following sense: for every chart (U i, ϕ i ), the map ω i : ϕ i (U i ) Alt p (R m ) defined by ω i (y) = Alt p (T y ϕ 1 i )(ω ϕ 1 ) is differentiable. Lemma 10.2. To check the above differentiability property at x M, it suffices to verify for one U i containing x. In particular, the differentiability condition depends only on the differentiable structure of M, and not on the specific atlas {(U i, ϕ i )} i I used. Definition 10.3. The vector space of all differential forms of degree p on M is denoted by Ω p (M). Example 10.4 (Degree 0). A differential form ω Ω 0 (M) is just a differentiable function ω: M R since Alt 0 (T x M) = R. Example 10.5 (Degree 1). If f: M R is a differentiable function, its differential is the form df Ω 1 (M) defined by the property that df(x) = T x f: T x M R for every x M. (Recall that Alt 1 (T x M) is just the dual of T x M.) 2. The wedge product Definition 10.6. The wedge product or exterior product of the differential forms α Ω p (M) and β Ω q (M) is the differential form α β Ω p+q (M) defined at each x M by (α β)(x) = α(x) β(x) Alt p+q (T x M). and Proposition 10.7. If α Ω p (M), β Ω q (M) and γ Ω r (M), then α (β γ) = (α β) γ β α = ( 1) pq α β. 3. Homomorphisms induced by maps Let f : M N be a differentiable between manifolds. For every x M, the tangent map T x f : T x M T f(x) N induces a map Alt p (T x f) : Alt p (T f(x) N) Alt p (T x M). We can therefore define a map Ω p (f): Ω p (N) Ω p (M) 31 i (y)

32 10. DIFFERENTIAL FORMS ON MANIFOLDS by the formula Ω p (f)(ω)(x) = Alt p (T x f)(ω(f(x)) for every ω Ω p (N) and every x M. The differential form Ω p (f)(ω) is often called the pull back of ω under the map f. Proposition 10.8. If f: M N, α Ω p (N) and β Ω p (N), then Ω p+q (f)(α β) = Ω p (α) Ω q (β). Proposition 10.9. If we have a composition of differentiable maps M f N g P, then Ω p (g f) = Ω p (f) Ω p (g). 4. Differential forms in local coordinates Let {(U i, ϕ i )} i I be an atlas for the manifold M. If ω Ω p (M) is a p form in M, the pull back ω i = Ω p (ϕ 1 i ) Ω p (V i ) is a differential form on the open subset V i = ϕ i (U i ) of R m. Proposition 10.10. There is a one-to-one correspondence between differential forms ω Ω p (M) and families of differential forms {ω i Ω p (V i )} i I that are compatible under coordinate changes ϕ j ϕ 1 i : ϕ i (U i U j ) ϕ j (U i U j ), in the sense that ω i ϕi(ui U j = Ω p (ϕ j ϕ 1 i )(ω j ϕj(u i U j ) for every i, j I. 5. The exterior derivative Theorem 10.11. As M ranges over all manifolds and p over all integers, there is a unique family of maps d: Ω p (M) Ω p+1 (M) such that: (1) d is linear; (2) if α Ω p (M) and β Ω q (M), d(α β) = (dα) β + ( 1) p α dβ; (3) if f: M N is differentiable, d Ω p (f) = Ω p+1 (f) d; (4) d d = 0; (5) if f Ω 0 (M) is interpreted as a function f: M R, then df Ω 1 (M) is defined by the property that df(x) = T x M at each x M. Definition 10.12. The form dω Ω p+1 (M) is the exterior derivative of ω. In practice, if {(U i, ϕ i )} i I is an atlas for M and if ω Ω p (M) is determined by the family {ω i Ω p (V i )} i I, then dω Ω p+1 (M) is determined by the family {dω i Ω p+1 (V i )} i I. The fact that the dω i are compatible with the coordinate changes ϕ j ϕi 1 comes from the fact that d commutes (up to degree shift) with the induced homomorphisms Ω p (ϕ j ϕ 1 i ).

CHAPTER 11 de Rham cohomology 1. De Rham cohomology Definition 11.1. A differential form ω Ω p (M) is closed if dω = 0. It is exact if there exists a form α Ω p 1 (M) such that ω = dα. Example 11.2. If U is an open subset of R 2, every differential form ω Ω 1 (U) can be written as ω = P dx + Q dy for some functions P, Q: U R. Then ω is closed if and only if P. If is exact if and only if there exists a function f : U R such that f x y = Q x = P and f y = Q. Since d d = 0, every exact form is closed. In other words, in Ω p (M), the kernel Z p (M) of d: Ω p (M) Ω p+1 (M) contains the image B p (M) of d: Ω p 1 (M) Ω p (M). Definition 11.3. The p th de Rham cohomology space of M is the vector space H p (M) quotient of Z p (M) by B p (M). In other words, the size of H p (M) measures the difference between closed and exact for p forms in M. Proposition 11.4. Let {M i } i I be the (finite or infinite) set of connected components of the manifold M. Then, for every p, H p (M) = i I H p (M i ). Recall that a form of degree 0 on a manifold M is the same as a function f : M R. This form is closed if and only if f is locally constant. In particular, considering the value of the constant: Proposition 11.5. If the manifold is connected, then H 0 (M) is canonically isomorphic to 0. 2. Homomorphisms induced by differentiable maps Consider a differentiable map f: M N. Because d Ω q (f) = Ω q+1 (f) d for q = p 1 and p, the induced map Ω p (f): Ω p (N) Ω p (M) sends Z p (N) to Z p (M) and B p (N) to B p (M). As a consequence, it induces a linear map H p (f): H p (N) H p (M) Proposition 11.6. Given two maps M f N g P, H p (g f) = H p (f) H p (g). Corollary 11.7. If f : M N is a diffeomorphism, then H p (f): H p (N) H p (M) for every p. 33

34 11. DE RHAM COHOMOLOGY 3. Invariance under homotopy Definition 11.8. Two differentiable maps f 0 : M N and f 1 : M N are (differentiably) homotopic if there exists a differentiable map H: M R N such that (1) H(x, 0) = f 0 (x) for every x M; (2) H(x, 1) = f 1 (x) for every x M. The definition is perhaps easier to understand if, for every t R, one considers the map f t : M N defined by f t (x) = H(x, t). The map H is called a homotopy from f 0 to f 1. Theorem 11.9. If f 0 : M N and f 1 : M N are homotopic, then the induced homomorphisms H p (f 0 ): H p (N) H p (M) and H p (f 1 ): H p (N) H p (M) are equal. The key step in the proof is the following lemma. Lemma 11.10. There exists for every p a homomorphism K p : Ω p (N) Ω p 1 (M) such that d K p + K p+1 d = Ω p (f 1 ) Ω p (f 0 ) Definition 11.11. Two manifolds M and N are homotopically equivalent if there exists two maps f: M N and g: N M such that g f is homotopic to the identity map of M, and f g is homotopic to the identity of N. Note that M and N may have different dimensions. Corollary 11.12. If M and N are homotopically equivalent, then H p (M) is isomorphic to H p (N) for every p. Example 11.13. An open subset U of R n is star-shaped if there exists x 0 U such that, for every x U, the line segment joining x 0 to x is completely contained in U. For instance, R n and any open ball in R n are star-shaped. If U is starshaped, it is homotopically equivalent to the 0 dimensional manifold {x 0 }. As a consequence, H p (U) = 0 if p 0, and H 0 (U) = R. Example 11.14. The space R n {0} is homotopically equivalent to the sphere S n 1 by consideration of the inclusion map f : S n 1 R n {0} and of the map g : R n {0} S n 1 defined by g(x) = x/ x. As a consequence, H p (R n {0}) is isomorphic to H p (S n 1 ) for every p. At this point, the only corollary we get out of this is that H n (R n {0}) = 0. 4. De Rham cohomology and wedge product Theorem 11.15. There exists a well-defined bilinear homomoprhism H p (M) H q (M) H p+q (M) which to [α] H p (M) and [β] H q (M) associates [α β] H p+q (M).