Notes 1: Review of Riemannian geometry. r @ k @ The Christoffel symbols are the components of the Levi-Civita connection: @ @xj = Γij @xk . ( ) @xi k 1 k` @ @ − @ Formula: Γij = 2 g @xi gj` + @xj gi` @x` gij . @ Exercise 1 Prove that if g (s) is a 1-parameter family of metrics with @s gij = vij (v is a symmetric covariant 2-tensor), then @ 1 Γk = gk` (r v + r v − r v ) : (1) @s ij 2 i j` j i` ` ij 2 M f ig @ Hint: Given p , let x be normal coordinates centered at p. Then @xi gjk (p) = 0. Compute in these coordinates. Use also the fact that the difference of two connections is a tensor and hence the variation of a family of connections is a tensor. Thus both sides of (1) are the components of tensors. Let α be a covariant r-tensor and X be a vector field. Lie derivative of α with respect to X is denoted by LX α. Formula: (LX α)(Y1;:::;Yr) = (rX α)(Y1;:::;Yr) (2) Xn r + α (Y1;:::;Yi−1; Yi X; Yi+1;:::;Yr) : i=1 Exercise 2 1. Using (2), show that the Lie derivative of the metric is given by L r r ( X g)(Y1;Y2) = g ( Y1 X; Y2) + g (Y1; Y2 X) : (3) 2. (Local coordinates) Show that if α is a covariant 2-tensor, then L r k` r r ( X α)ij = X αij + g ( iXkα`j + jXkαi`) and in particular L r r ( X g)ij = iXj + jXi; so that if f is a function, then ( ) Lgrad f g = 2rirjf: (4) g ij m Let Rijk` = g`mRijk be the components of the Riemann curvature 4-tensor. The first and second Bianchi identities are Rijk` + Rjki` + Rkij` = 0; (5) riRjk`m + rjRki`m + rkRij`m = 0: (6) Exercise 3 Let Rij be the Ricci tensor and let R be the scalar curvature. 1. Prove the contracted second Bianchi identity: ij 2g riRjk = rkR: (7) 1 1 ≥ 2. Using (7), show that if g is an Einstein metric, i.e., Rij = n Rgij, and n 3; then R is a constant. The Ricci identities: Xr m (r r − r r ) α ··· = − R α ··· ··· : (8) i j j i k1 kr ijk` k1 k`−1mk`+1 kr `=1 Exercise 4 Prove: 1. if α is a 1-form, then r r − r r − ` ( i j j i) αk = Rijkα`: 2. If β is a 2-tensor, then r r − r r − p − p i jβk` j iβk` = Rijk βp` Rij` βkp: (9) ij ∆ = g rirj is the Laplacian. Exercise 5 Prove that for any function f 1. ∆rif = ri∆f + Rijrjf: (10) 2. 2 2 ∆ jrfj = 2 jrrfj + 2Rijrifrjf + 2rifri (∆f) : (11) 3. Prove that if Rc ≥ 0, ∆f ≡ 0, and jrfj = 1; then r2f = r(rf) = 0, i.e., rf is parallel, and Rc (rf; rf) = 0. Divergence theorem: Let (Mn; g) be a compact oriented Riemannian manifold. Let dµ denote the volume form of (M; g), let dσ denote the area form of @M, and let ν denote the unit outward normal to @M. If X is a vector field, then Z Z div (X) dµ = hX; νi dσ; (12) M @M R r i M where div (X) = iX . In particular, if is closed, then M div (X) dµ = 0. Exercise 6 Prove: R 1. On a closed manifold, M ∆udµ = 0: 2. On a compact manifold, Z Z ( ) @v @u (u∆v − v∆u) dµ = u − v dσ: M @M @ν @ν In particular, on a closed manifold Z Z u∆vdµ = v∆udµ. M M 2 3. Show that if f is a function and X is a 1-form, then Z Z Z f div (X) dµ = − hrf; Xi dµ + f hX; νi dσ: M M @M Exercise 7 By choosing coordinates where gij = δij at a point, show that for any 2-tensor aij 1 ( ) ja j2 ≥ gija 2 : ij g n ij More generally, for a p-tensor a, p ≥ 2, 2 1 ij 2 a ··· ≥ g a ··· : k1 kp n ijk3 kp Let Rc denote the Ricci tensor. Exercise 8 Show that on a closed manifold Z Z Z r2f 2 dµ + Rc (rf; rf) dµ = (∆f)2 dµ. (13) M M n M jr2 j2 ≥ 1 2 Since f n (∆f) ; this implies Z Z n − 1 Rc (rf; rf) dµ ≤ (∆f)2 dµ. (14) M n M Exercise 9 (Lichnerowicz inequality) Suppose f is an eigenfunction of the Laplacian with eigen- value λ > 0: ∆f + λf = 0: Use (14) to show that if Rc ≥ (n − 1) K; where K > 0 is a constant, then λ ≥ nK: (15) @ Exercise 10 Show that if @s gij = vij; then ( ) r r v + r r v − r r v @ ` 1 `p i j kp i k jp i p jk Rijk = g (16) @s 2 −∇jrivkp − rjrkvip + rjrpvik 1 = g`p (r r v − r r v − r r v + r r v ) (17) 2 i k jp i p jk j k ip j p ik 1 − g`p (R v + R v ) : 2 ijkq qp ijpq kq @ Exercise 11 Show that if @s gij = vij; then: 1. ( ) @ 1 R = − ∆ v + r r V − r (div v) − r (div v) ; @s ij 2 L ij i j i j j i where the Lichnerowicz Laplacian is defined by ∆Lvij + ∆vij + 2Rkij`vk` − Rikvjk − Rjkvik: (18) 3 2. @ R = r r v − ∆V − v · R : @s ` i i` ij ij @ − We say that a 1-parameter family of metrics g(t) is a solution to the Ricci flow if @t gij = 2Rij. Exercise 12 Show that if g(t) is a solution to the Ricci flow, then 1. @R = ∆R + 2 jRcj2 : @t 2. @ R = ∆ R = ∆R + 2R R − 2R R : @t ij L ij ij kij` k` ik jk 3. @ R = ∆R + 2 (B − B + B − B ) (19) @t ijk` ijk` ijk` ij`k ikj` i`jk − (RipRpjk` + RjpRipk` + RkpRijp` + R`pRijkp) ; where pr qs Bijk` + −g g RipjqRkr`s = −RpijqRq`kp: (20) 4.