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Do Carmo Riemannian Geometry 1. Review Example 1.1. When M

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Do Carmo Riemannian Geometry 1. Review Example 1.1. When M DIFFERENTIAL GEOMETRY NOTES HAO (BILLY) LEE Abstract. These are notes I took in class, taught by Professor Andre Neves. I claim no credit to the originality of the contents of these notes. Nor do I claim that they are without errors, nor readable. Reference: Do Carmo Riemannian Geometry 1. Review 2 Example 1.1. When M = (x; jxj) 2 R : x 2 R , we have one chart φ : R ! M by φ(x) = (x; jxj). This is bad, because there’s no tangent space at (0; 0). Therefore, we require that our collection of charts is maximal. Really though, when we extend this to a maximal completion, it has to be compatible with φ, but the map F : M ! R2 is not smooth. Definition 1.2. F : M ! N smooth manifolds is differentiable at p 2 M, if given a chart φ of p and of f(p), then −1 ◦ F ◦ φ is differentiable. Given p 2 M, let Dp be the set of functions f : M ! R that are differentiable at p. Given α :(−, ) ! M with 0 α(0) = p and α differentiable at 0, we set α (0) : Dp ! R by d α0(0)(f) = (f ◦ α) (0): dt Then 0 TpM = fα (0) : Dp ! R : α is a curve with α(0) = p and diff at 0g : d This is a finite dimensional vector space. Let φ be a chart, and αi(t) = φ(tei), then the derivative at 0 is just , and dxi claim that this forms a basis. For F : M ! N differentiable, define dFp : TpM ! TF (p)N by 0 0 (dFp)(α (0))(f) = (f ◦ F ◦ α) (0): Need to check that this is well-defined. n k −1 −1 Example 1.3. If F : R ! R with k ≤ n, and dFα is surjective for all x 2 F (0), then F (0) is a smooth manifold. TM = f(x; V ): X 2 M; v 2 TpMg. M smooth manifold, G a group acting on M properly discontinuously, that is, F : G × M ! M so that for all g 2 G, Fg : M ! M by x 7! F (g; x) is a diffeomorphism. Fgh = FgFh and Fe = 1. For all x 2 M, there exists U neighbourhood n of x such that Fg(U) \ U = ; for all g 2 G, g 6= e. Set N = M = fx ∼ Fg(x): g 2 Gg is a manifold. Recall: F : M ! M is a diffeo if F bijective, differentiable, AND F −1 is diff Theorem 1.4. Given M smooth manifold, there is M~ simply connected manifold such that π1(M) acts properly discon- n tinuously on M~ and M = M/π~ 1(M). Example 1.5. RPn = Sn= fx ∼ −xg. CPn = Cn+1 − f0g= fx ∼ λx : λ 6= 0g. Definition 1.6. A vector field X is a map M ! TM. Let X(M) be the set of all vector fields. Given X; Y 2 X(M), define the Lie bracket to act by [X; Y ](f) = X (Y (f)) − Y (X(f)) : 1 DIFFERENTIAL GEOMETRY NOTES 2 Satisfying [[X; Y ] ;Z] + [[Y; Z] ;X] + [[Z; X] ;Y ] = 0: 2. Riemannian Metric n Definition 2.1. Let M be an n-dimensional manifold. A metric g on M is a symmetric bilinear map gp : TpM×TpM ! R for all p 2 M so that (1) gp (aX + Y; Z) = agp (X; Z) + gp (Y; Z) all a 2 R, X; Y; Z 2 TpM (2) gp (X; Y ) = gp (Y; X) (3) gp(X; X) > 0 for all X 2 TpM − 0 (4) If X; Y 2 X(M), then p 7! gp (X(p);Y (p)) is a smooth function Theorem 2.2. Every manifold has a metric (partition of unity) n @ @ Let (U; ') be a chart for M . Then gij(x) = g ( ; ) for all x 2 U and i; j = 1; :::; n. Then the matrix (gij) is '(x) @xi @xj positive definite and symmetric. Additionally, gij(x) are smooth functions of x. P @ P @ If we write X; Y 2 T M as X = ai and Y = bj then '(x) @xi @xj X @ @ X g (X; Y ) = a b g ; = a b g : '(x) i j ' @x @x i j ij i;j i j i;j ' −1 (V; ') is another chart, gij from first chart, gij from another chart. Let h = ◦ φ, then Claim 2.3. @ = P @hk @ . @xi k @xi @yk Proof. For all f 2 C1(M), @ @ @ X @hk @f ◦ (f) = (f ◦ φ) = (f ◦ ◦ h) = ◦ : @xi @xi @xi @xi @yk Therefore, φ @ @ X @hk @h` gij = g ; = gk`: @xi @xj @xi @xj k;` As a matrix, φ T gij = (Dh) g (Dh) : Given a curve γ : I ! M, the length(γ) = pg (γ0(t); γ0(t))dt: ˆI If R is a region of M, so that R ⊆ φ(U) of a chart, q vol(R) = det(gij)dx1:::dxn: ˆφ−1(R) If R is not contained in a single chart, use a partition of unity. n n n Example 2.4. On R , for X; Y 2 TxR = R , gx(X; Y ) = X · Y is called the Eucliean metric. n n+k If M ⊆ R is a manifold, gp (X; Y ) = X · Y is called induced metric. n n+1 n+1 n S = x 2 R : jxj = 1 ⊆ R then gsn is the induced metric on S . n n 4 B = fx 2 : jxj < 1g then g n (X; Y ) = X · Y is called the hyperbolic metric R H 1−|xj22 Definition 2.5. F :(M; g) ! (N; h) is an isometry if (1) F is a local diffeomorphism (2) g(X; Y ) = h ((dF )(X); (dF )(Y )) for all X; Y 2 X(M) DIFFERENTIAL GEOMETRY NOTES 3 Example 2.6. Any two curves of the same length are isometric. f(x; y): y > 0g is isometric to a cone (locally) Isometries of (Rn; Euclidean Metric) are rigid motions n Isometries of (S ; gsn ) = O(n + 1). For A 2 O(n + 1), Ax · Ax = AT A x · x = x · x = 1 for all x 2 Sn: Therefore, A is an isometry of Sn. n n n n n n Isometries of (H ; gH ) are comformal maps from B to B . F : B ! B is comformal if (dF )x (X) · (dF )x (Y ) = λ(x)X · Y . Definition 2.7. A connection r : X(M) × X(M) ! X(M) so that 1 (1) rfX+Y Z = frX Z + rY Z where f 2 C (M) (2) rX (fY + Z) = X(f)Y + frX Y + rX Z A connection is symmetric if rX Y − rY X = [X; Y ] : A connection is compatible with metric g if for all X; Y; Z 2 X(M), X (g(Y; Z)) = g (rX Y; Z) + g (Y; rX Z) : A connection compatible with g and symmetric is called a Levi-Civita connection. In coordinates (U; '), @ X k @ r @ = Γij ; @xi @xj @xk k P @ P @ where Γ is called the Christoffel symbols. If X = ai and Y = bj , then @xi j @xj 0 1 X X @ X @ X @bj @ rX Y = air @ @ bj A = aibjr @ + ai @xi @x @xi @x @x @x i j j i;j j i;j i j X k @ X @ = aibjΓi;j + X(bj) @xk @xj i;j;k j X k @ X @ = aibjΓij + X(bk) @xk @xk i;j;k k Suppose γ : I ! M is a curve, and Y (t) 2 Tγ(t)M. If γ has self intersection, γ(t1) = γ(t2), Y may take different values there. rγ0(t)Y is well-defined from the above expression, because the first term just need values. The second part, we just need that the first vector field X to be differentiable along the integral curves defined by Y . Theorem 2.8. Levi-Civita connections exist and are unique. Proof. We have X (hY; Zi) = hrX Y; Zi + hY; rX Zi Y (hZ; Xi) = hrY Z; Xi + hZ; rY Xi Z (hX; Y i) = hrZ X; Y i + hX; rZ Y i Now add the first two, and subtract the last, to get (using symmetricity) X (hY; Zi) + Y (hZ; Xi) − Z (hX; Y i) = 2 hrX Y; Zi + hZ; [Y; X]i + hY; [X; Z]i + hX; [Y; Z]i : Therefore, 1 hr Y; Zi = (X hY; Zi + Y hZ; Xi − Z hX; Y i) − hZ; [Y; X]i − hY; [X; Z]i − hX; [Y; Z]i ; X 2 DIFFERENTIAL GEOMETRY NOTES 4 which defines it. For a chart (U; '), then by combining symmetry and Christofell symbols, @ @ X ` 1 @ @ @ r @ ; = Γij · gk` = gjk + gik − gij : @xi @xj @xk 2 @xi @xj @xk ` If gij is g−1 , then ij ij k` ` X g @ @ @ Γij = gjk + gik − gij : 2 @xi @xj @xk k n Example 2.9. On (R ; Euclidean) ;DX Y = (X(y1); :::; X(yn)) = (hX; Dy1i ; :::; hX; Dyni) where Y = (y1; :::; yn). For M n ⊆ Rn+k, X; Y 2 X(M), T rX Y = (DX Y ) induced connection (the T is the projection to the tangent space, as a subspace of Rn+k). 0 Definition 2.10. Let (M; g) be a Riemannian manifold. A curve γ : I ! M is a geodesic if rγ0(t)γ (t) = 0 for all t 2 I (its acceleration is constant). n 0 0 0 Example 2.11. On R with the Euclidean metric, a curve γ(t) = (γ1(t); :::; γn(t)) and so γ (t) = (γ1(t); :::; γn(t)). Then 0 0 00 00 00 rγ0(t)γ (t) = Dγ0(t)γ (t) = (γ1 (t); :::; γn(t)) = γ (t): Therefore, γ ⊆ Rn is a geodesic iff γ00(t) = 0 for all t.
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