DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES MAGGIE MILLER September 25, 2015 1. 09/16/2015 1.1. Textbooks. Textbooks relevant to this class are Riemannian Geometry by do Carmo Riemannian Geometry by Petersen Lectures on Differential Geometry by Schoen and Yau Riemannian Geometry by Jost. Lectures on Differential Geometry has several problem sets at the end helpful for struggling students. 1.2. Basic Definitions. Definition 1.1. An n-dimensional smooth atlas on a topological space M n (which is Hausdorff and second countable is a collection of charts x α (α Λ, where each x α : U α R n V α M n is a homeomorphism of open sets so that (1 α Λ V α = M. (2 If x α (U α X β (U β = W, then x 1 β X α : x 1 α (W x 1 (W is a diffeomorphism. β Definition 1.2. An n-dimensional smooth manifold is some topological space M n endowed with a maximal (with respect to properties (1 and (2 n-dimensional atlas. Remark 1.3. Every atlas is contained in a unique maximal atlas. Example 1.4. The identity map id R n : R n R n gives an atlas on R n. This atlas in contained the maximal atlas {f : U V f is a diffeomorphism and U, V are open subsets of R n }, the usual smooth structure on R n. Remark 1.5. A manifold has a countable atlas if and only if it is second countable. Definition 1.6. A continuous map f : M m N n is differentiable (smooth if for every p M there exist charts x α, y β in the charts for M m, N n respectively so that p x α (U α, f(p y α U β, and y 1 β f x α is smooth where defined. Definition 1.7. A map f : M m N n is a diffeomorphism if it is smooth and has a smooth inverse f 1 : N n M m. Definition 1.8. A set X M m is an n-dimensional submanifold of M m if for each p X, there exists an open neighborhood U of p in M m and a diffeomorphism φ : U R m so that φ(u X = R n. Definition 1.9. Given a function f : U R n R m on open set U, we say that a R m is a regular value of f if for all p f 1 (a, df(p : R n R m is surjective. If a is a regular value, then f 1 (a is a submanifold of dimension n m. Example 1.10. Submanifolds of R n gain a natural atlas from R n. In particular, the pre-image of a regular value of any function on R n has a natural atlas. Theorem 1.11. [Weak Whitney s Embedding Theorem] Every n-dimensional manifold M n is diffeomorphic to a submanifold of R 2n+1. 1
2 MAGGIE MILLER Example 1.12. [Real Projective Space] The space RP n is the space of all lines in R n+1 containing the origin. Given a vector v R n+1 \{0}, let [v] denote the line containing v, i.e. [v] = {λv λ R}. Then the following is an atlas on RP n : φ i : R n V i RP n (i = 1,..., n + 1 (x 1,..., x n [(x 1,..., x i 1, 1, x i,..., x n ]. Example 1.13. [Complex Projective Space] The space CP n is the space of all complex dimension-1 subspaces of C n+1. Note CP n is a 2n-dimensional real manifold. Given a vector v R n+1 \{0}, let [v] denote the line containing v, i.e. [v] = {λv λ C}. Then the following is an atlas on CP n : φ i : C n V i CP n (i = 1,..., n + 1 (x 1,..., x n [(x 1,..., x i 1, 1, x i,..., x n ]. In this example, the transition maps x 1 i x j are not only diffeomorphisms, but holomorphic when we view CP n as an n-dimensional complex manifold. Example 1.14. Given atlases x α (α Λ, y β (β B on M m, N n respectively we have a natural atlas on M m N n gives by x α y β ((α, β Λ B. Example 1.15. [Quotient spaces] Quotient spaces gain an atlas from the projection map. As an example of a quotient space, consider the n-dimensional torus R n / where (x 1,..., x n (y 1,..., y n if there exists n i Z so that y i = x i + n i. Definition 1.16. A left group action of a group G on a manifold M is a map φ : G M M (g, p φ g (p = gp such that (1 φ g : M M is a diffeomorphism, (2 φ i = id M, (3 φ g1g 2 = φ g1 φ g2. We say that G acts on M and may write G M. We define a quotient by the group action as M/G = {[p] p M}, where [p] = {q there exists g G such that gp = q} is the orbit of p M. Remark 1.17. The n-dimensional torus R n / is given by R n /Z n, where Z n R n by φ : Z n R n R n ((m 1,..., m n, (x 1,..., x n (x 1 + m 1,..., x n + m n Definition 1.18. A group action φ : G M M is properly discontinuous if for all p M there exists an open neighborhood U M of p such that φ g (U U = for all g e. Proposition 1.19. Suppose G M properly discontinuously. Then there exists a natural atlas on M/G such that projection π : M M/G is smooth. Remark 1.20. Z 2 S n where φ 0 is the identity and φ 1 is the antipodal map. Then S n /Z 2 = RP n. Definition 1.21. A Lie group G is a group and an n-manifold such that the maps are both smooth. m : G G G (g, h gh i : G G g g 1 Example 1.22. R n is a Lie group with the group operation of coordinatewise addition. GL(n, R, SL(n, R, O(n are Lie groups with their usual structure. GL(n, R has manifold structure as an open subset (and therefore submanifold of R n2. SL(n, R, O(n have manifold structures as the preimages of a regular value (SL(n, R = f 1 (1, O(n = g 1 (I n where f(a = det A, g(a = AA T.
DIFFERENTIAL GEOMETRY 3 Definition 1.23. Given g in a group G, left translation by g is the map L g : G G h gh. We may similarly define right translation by g, R g : G G h hg. Remark 1.24. L g is a diffeomorphism, with L 1 g = L g 1. Similar holds for R g. 1.3. Closing Remarks. For any closed subset C R n, R n \C is an n-dimensional manifold. Thus, it is hard to classify manifolds in general. Generally one focuses on classifying compact n-manifolds (up to diffeomorphism. S 1 is the only compact 1-manifold. The complete set of compact 2-manifolds is {S 2, RP 2 } {Σ g g N} {RP 2 #Σ g g N}, where Σ g is the orientable surface of genus g and # refers to connect sum. By Thurston s generalization, we cannot give a list of compact 3-manifolds but we may uniquely describe a compact 3-manifold as the connect sum of a finite number of prime 3-manifolds. In higher dimensions, we must distinguish between manifolds that are homeomorphic and diffeomorphic. Milford found many exotic 7-spheres, i.e. distinct smooth structures on S 7. The following question remains open. Question 1.25. If M 4 is homeomorphic to S 4, must M 4 be diffeomorphic to S 4? 2.1. Tangent Spaces. 2. 09/21/2015 Definition 2.1. Let M n be an n-dimensional differentiable manifold and α : (, M n be a smooth map. Let p = α(0. Given local coordiantes x near p, write α(t = x(x 1 (t,..., x n (t. Then the tangent vector to α at p is given by α (0 = ( d dt x 1(0,..., d dt x n(0. Remark 2.2. Let f be a differentiable function defined in some neighborhood of p. Then α (0 f = d dt (f α. That is, d dt f(α(t n (f x = (x t=0 x i(0. i Definition 2.3. The tangent space of M at p is the collection T p M of all tangent vectors t p. Remark 2.4. T p M has a well-defined structure of an n-dimensional vector space. i=1 Definition 2.5. Let x : U R n V M be local coordinates near some p V. The coordinate basis { x 1,..., x n } T p M is defined by x i = ( x i (0, where x i (t = x(x 0 + te i. In coordinates we can write α (0 = x i (0 x i. Definition 2.6. Let f : M m N n be a differentiable map between manifolds. The derivative of f is the linear map given by (df p : T p M T f(pn α (0 (f α (0. Definition 2.7. The tangent bundle of M n is given by T M = {(p, v p M, v T p (M}. Proposition 2.8. T M is a differentiable manifold of dimension 2n.
4 MAGGIE MILLER Proof. Given a chart x : U R n V M in an atlas of M, define to otbain charts forming an atlas on T M. ȳ : U R n W T M (x, y ( y i x i x(x Definition 2.9. We say M is orientable if there exists an atlas {U α, x α } of M such that changes of coordinates preserve orientation (i.e. det d(x 1 β x α > 0 where it makes sense. An orientation on M naturally gives an orientation on each T p M. 2.2. Vector Fields. Definition 2.10. A vector field on U M is a map X : U T M p T p (M so that local expressions are smooth. In coordinates, we write X(p = a i (p x i where each a i is smooth. Definition 2.11. Let X be a vector field on U. For each p U, there exists a neighborhood W U of p and a differentiable map φ : ( ɛ, ɛ W M (t, q φ(t, q := φ t (q such that t φ t (q is the unique solution to the ordinary differential equation that {φ t } is a flow for X. { α t = X(α(t α(0 = q. We say Definition 2.12. Given vector fields X, Y on M, [X, Y ] := XY Y X is called the bracket of X and Y. Proposition 2.13. [X, Y ] is a vector field. Proof. In coordinates write X = a i x i, Y = b j y j. Then XY Y X = ( a i bj ( b j ai x i x j x j x i = b j a i a i b j. x i,j i x j y i,j j x i Remark 2.14. Note [ x i, x j ] = 0, but generally [X, Y ] may not be zero. Definition 2.15. Let φ : M m N n be a differentiable map, and X, X be vector fields on M m, N n respectively. We say that X, X are φ-related if for all p M, X(φ(p = dφp (X(p. Remark 2.16. The bracket has the following properties. (1 [X, Y ] = [Y, X] (antisymmety. (2 [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0 (Jacobi identity. (3 [X, Y ] 0 if and only if local flows of X, Y commute. (4 If X, X are φ-related and Y, Ȳ are φ-related, then [X, Y ], [ X, Ȳ ] are φ-related. Note that properties 1 and 2 define a Lie algebra.
DIFFERENTIAL GEOMETRY 5 2.3. Tensors. Definition 2.17. A tensor of type (r, s on M is some multilinear map p M T : r i=1 T p M s j=1 T pm R. Example 2.18. A vector field X is a tensor of type (1, 0. X defines a vector at p, and given w T p M, X(ω = ω(x R. Example 2.19. A differential form α of degree k is a tensor of type 0, k. For any v 1,..., v k T p M, α(v 1,..., v k R. 2.4. The Lie Derivative. Definition 2.20. Let T be a tensor of type (0, s on N. Let φ : M N be differentiable. We define the pullback of T along φ to be the tensor φ T of type (0, s on M given by φ T (v 1,..., v s = T (dφ(v 1,... dφ(v s. Definition 2.21. Let X be a vector field with local flow {φ t }. The Lie derivative of X is given by d dt φ t (T t=0 := L X T. 2.5. Implicit and Inverse Function Theorems. Definition 2.22. A differentiable map φ : M m N n is called an immersion if dφ p : T p M T φ(p N is injective for all p M. Theorem 2.23. [Inverse Function Theorem] Let φ : M N be a differentiable map so that dφ p : T p M T φ(p N is an isomorphism. Then there exist neighborhoods U M of p and V N of φ(p such that φ : U V is a diffeomorphism. Theorem 2.24. [Implicit Function Theorem] Let φ : R n+m R n be a differentiable map and fix coordinates ( x, ȳ for R n+m. Suppose φ(p, q = c. If the matrix [fracφ i ȳ j (p, q] is invertible, then there exist open neighborhoods U R n of p and V R m of q so that there exists a unique differentiable g : U V with {(z, g(z z U} = {(z, y U V f(z, y = c}. Remark 2.25. Theorems 2.23 and 2.24 are equivalent. 2.6. Partitions of Unity. Definition 2.26. Given open covers {U α }, {V β } for M, we say that {V β } is a refinement of {U α } if for each β there exists some α := α(β so that V β U α(beta. Definition 2.27. Given an open cover U α of M, we say that {U α } is locally finite if for each p M there exists a neighborhood V of p so that V U α = for all but finitely many α. Remark 2.28. Every open cover of a manifold M admits a locally finite refinement. Definition 2.29. A partition of unity subordinated to a locally finite cover {U α } is a collection of differentiable functions φ α : M R so that (1 supp(φ α U α (2 0 φ α 1 (3 α Λ φ α(x = 1. Note that φ α (x = 0 for all but finitely many α. 2.7. Riemannian Metrics. Definition 2.30. A smooth Riemannian metric on a manifold M n is the type (0, 2 tensor p M scalar product g p : T p M T p M R, where g p is positive definite. In coordinates, we require g ij to be ( ( smooth, where g ij (x = x, i x x(x j. We refer to {g ij } as local coordinates of g. x(x Definition 2.31. Let T be a tensor of type (r, s on M. Let {dx 1,..., dx n } be a basis of Tp M. Then the ( components of T are T i1,...,ir j 1,..., j s := T dx i1,..., dx ir, x j1,..., x js.
6 MAGGIE MILLER Proposition 2.32. For any smooth manifold M n, there exists a Riemannian metric on M n. Proof. By Whitney s theorem, there exists an embedding (i.e. an immersion that is a homeomorphism onto its image φ : M n R 2n+1. Define X, Y M = dφ(x, dφ(y R 2n+1. Alternate. Let {V α } α Λ be a locally finite open cover of M such that V α = x α (U α for charts x α : U α R n M. For a point p V α, define g α (u, v ( = dx 1 α (u, dx 1 α (v R n. Then g a lpha is a Riemannian metric on V α with components g α ij = g α x i, x j = e i, e j R n = δ ij. Let {φ α } α Λ be a partition of unity suborindate to {V α } α Λ. Then define g p (u, v = α φ α(p(g α p (u, v. This sum is well-defined even though g α is not defined outside of V α since φ α vanishes outside of V α. Then g is a Riemannian metric on M (using the fact that α φ α(p = 1. Remark 2.33. Note that in both proofs of Proposition 2.32, the metric obtained on M is the pullback of the metric on φ(m = R m. 3.1. Theorema Egregium. 3. 09/23/2015 Definition 3.1. Given a surface S R 3, p S, and v T p S, let P v be the plane in R 3 containing p and parallel to v and the normal vector to S at p. Let C v = P v S, and Π(v be the curvature of C v at p. Then the principal curvatures of S at p are the eigenvalues of Π. Remark 3.2. The principal curvatures are given by κ 1 = min v =1 Π(v, κ 2 = max v =1 Π(v. Definition 3.3. We define mean curvature H p and the Gauus curvature K p of S at p by H p = κ 1 + κ 2, K p = κ 1 κ 2. 2 Theorem 3.4. [Theorema Egregium] The Gauus curvature K p is an intrinsic quality of S (i.e. does not depend on the embedding S R 3. 3.2. Length and Distance. Definition 3.5. Let (M n, g M, (N n, g N be Riemannian manifolds. A diffeomorphism φ : M N is an isometry f for all u, v T p M for each p M. g N (dφ(u, dφ(v = g M (u, v Definition 3.6. Let (M n, g be a Riemannian manifold and α : [a, b] M a piecewise differentiable curve. Then the length of α is given by b L(α := α (t α(t dx, where α (t α(t = g α(t (α (t, α (t. a Definition 3.7. Given a Riemannian manifold (M n, g, the distance between points p, q M is given by d g (p, q = inf{l(α α : [a, b] M piecewise differentiable and α(a = p, α(b = q}. Proposition 3.8. If (M, g is a Riemannian manifold, then (M, d g is a metric space. Proof. We must show that: (1 d g (p, q = 0 if and only if p = q, (2 d g (p, q 0 for all p, q M, (3 d g (p, r d g (p, q + d g (q, r for all p, q, r M.
DIFFERENTIAL GEOMETRY 7 Property (2 follows from the fact that L(α is non-negative for every piecewise differentiable α. To show property (3, let p, q, r M. Given ɛ > 0, choose curves α from p to q and β from q to r so that L(α < d g (p, q + ɛ, L(β < d g (q, r + ɛ. Then the concatenation of α, β (dentoed α + β is a curve from p to r and L(α + β = L(α + L(β, so d g (p, r L(α + β < d g (p, q + d g (q, r + 2ɛ. Property (3 follows. Finally, we show property (1 holds. Let p q M. Choose a chart x α : U α R n V α M so that (p. Choose ɛ > 0 so that q x α (B ɛ (x 0 U α. Let α be any piecewise differentiable curve from p to q, and define t = sup{t > α α([a, t] x α (B ɛ (x 0 }. Then α [a, t] : [a, t] x α (B α (x 0 is a piecewise differentiable curve with α(a = p and α( t x α (B ɛ (x 0. Recall g α (u, v = g(dx α (u, dx α (v, so there exists some constant C > 0 so that δ C g α Cδ, where δ denotes the Euclidean metric on R n. Then p V α. Let x 0 = x 1 α L g (α α, t = L gα ((x 1 α α [α, t] 1 L g ((x 1 α α [α, t] C ɛ C. Thus, d g (p, q ɛ C > 0. 3.3. Volume. Definition 3.9. Let (M n, g be an oriented Riemannian manifold. The volume form on M, denoted dvol, is a differential form of degree n with the property that when {e 1,..., e n } is a positive orthonormal basis for T p M, dvol(e 1,..., e n = 1. ( Remark 3.10. In coordinates, dvol = ψ(xdx (dx = dx 1 dx n, where ψ(x = dvol x 1,..., x n. Write x i = j a ije j so g ik =, x i x k = j a ij e j, l a kl e l = j a ij a kj g = aa T. Note Therefore, dvol(ae 1,..., Ae n = (det Advol(e 1,..., e n = det(a. ( dvol,..., x 1 = det a = det g. x n Definition 3.11. Let (M, g be a Riemannian manifold with charts x α : U α V α M. Given R x α (U α, define the volume of R to be vol(r = det gdx. 3.4. Examples. x 1 α (R Example 3.12. Flat (standard Euclidean space (R n, δ is a Riemannian manifold. The distance function d δ is the familiar linear distance. Example 3.13. The standard sphere (S n, ḡ, where ḡ is the restriction of δ to S n = {x x 2 = 1} R n+1 is a Riemannian manifold.
8 MAGGIE MILLER Example 3.14. On R n+1 define (x 1,..., x n+1, (y 1,..., y n+1 = n i=1 x iy i x n+1 y n+1. This is not a Riemannian metric, but is Riemannian when restricted to H n = {x x, x = 1, x n+1 > 0}. We refer to (H n,, as (standard hyperbolic space. Remark 3.15. The Riemannian manifolds of Examples 3.12, 3.13, 3.14 satisfy the following symmetry property: Given p, q M and orthonormal bases {e i } T p M, {ē i } T q M, there exists an isometry φ : M M so that φ(p = q and dφ p (e i = ē i. In some sense, these three examples have maximal symmetry. Example 3.16. Let φ : M m N n be an immersion where N n, g N is a Riemannian manifold. Define the pullback metric g M = φ (g N. Then M m, g M is a Riemannian manifold. In particular, we consider M m N n and φ the inclusion map to see that submanifolds of Riemannian manifolds are naturally Riemannian. Example 3.17. Let (M m, g M, (N n, g N be Riemannian manifolds and define (M N, g M N by g M N (u, v = g M (dπ M (u, dπ M (v + g N (dπ N (u, dπ N (v. Then g M N is a Riemannian metric on M N. Example 3.18. Let G be a Lie group with identity element e. Given, on T e G, define g x (u, v = dl 1 x (u, dl 1 x (v. Then g is a Riemannian metric on G and by construction, each L x : G G is an isometry. We say that the metric is left-invariant. Note that Lie groups have the following symmetry property: for any x, y G, there exists an isometry φ : G G such that φ(x = y (e.g. φ = L yx 1.