Notes 1: Review of Riemannian geometry.

∇ ∂ k ∂ The Christoffel symbols are the components of the Levi-Civita connection: ∂ ∂xj = Γij ∂xk . ( ) ∂xi k 1 kℓ ∂ ∂ − ∂ Formula: Γij = 2 g ∂xi gjℓ + ∂xj giℓ ∂xℓ 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 = gkℓ (∇ v + ∇ v − ∇ v ) . (1) ∂s ij 2 i jℓ j iℓ ℓ ij ∈ M { i} ∂ 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) = (∇X α)(Y1,...,Yr) (2) ∑n ∇ + 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 ∇ ∇ ( 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 ∇ kℓ ∇ ∇ ( X α)ij = X αij + g ( iXkαℓj + jXkαiℓ) and in particular L ∇ ∇ ( X g)ij = iXj + jXi, so that if f is a function, then ( )

Lgrad f g = 2∇i∇jf. (4) g ij

m Let Rijkℓ = gℓmRijk be the components of the Riemann curvature 4-tensor. The first and second Bianchi identities are

Rijkℓ + Rjkiℓ + Rkijℓ = 0, (5)

∇iRjkℓm + ∇jRkiℓm + ∇kRijℓm = 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 ∇iRjk = ∇kR. (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: ∑r m (∇ ∇ − ∇ ∇ ) α ··· = − R α ··· ··· . (8) i j j i k1 kr ijkℓ k1 kℓ−1mkℓ+1 kr ℓ=1 Exercise 4 Prove:

1. if α is a 1-form, then ∇ ∇ − ∇ ∇ − ℓ ( i j j i) αk = Rijkαℓ. 2. If β is a 2-tensor, then ∇ ∇ − ∇ ∇ − p − p i jβkℓ j iβkℓ = Rijk βpℓ Rijℓ βkp. (9)

ij ∆ = g ∇i∇j is the Laplacian.

Exercise 5 Prove that for any function f

1. ∆∇if = ∇i∆f + Rij∇jf. (10) 2. 2 2 ∆ |∇f| = 2 |∇∇f| + 2Rij∇if∇jf + 2∇if∇i (∆f) . (11) 3. Prove that if Rc ≥ 0, ∆f ≡ 0, and |∇f| = 1, then ∇2f = ∇(∇f) = 0, i.e., ∇f is parallel, and Rc (∇f, ∇f) = 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 ∫ ∫ div (X) dµ = ⟨X, ν⟩ dσ, (12) M ∂M ∫ ∇ i M where div (X) = iX . In particular, if is closed, then M div (X) dµ = 0. Exercise 6 Prove: ∫ 1. On a closed manifold, M ∆udµ = 0. 2. On a compact manifold, ∫ ∫ ( ) ∂v ∂u (u∆v − v∆u) dµ = u − v dσ. M ∂M ∂ν ∂ν In particular, on a closed manifold ∫ ∫ u∆vdµ = v∆udµ. M M

2 3. Show that if f is a function and X is a 1-form, then ∫ ∫ ∫ f div (X) dµ = − ⟨∇f, X⟩ dµ + f ⟨X, ν⟩ dσ. M M ∂M

Exercise 7 By choosing coordinates where gij = δij at a point, show that for any 2-tensor aij 1 ( ) |a |2 ≥ 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 ∫ ∫ ∫

∇2f 2 dµ + Rc (∇f, ∇f) dµ = (∆f)2 dµ. (13) M M n M |∇2 |2 ≥ 1 2 Since f n (∆f) , this implies ∫ ∫ n − 1 Rc (∇f, ∇f) 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 { } ∇ ∇ v + ∇ ∇ v − ∇ ∇ v ∂ ℓ 1 ℓp i j kp i k jp i p jk Rijk = g (16) ∂s 2 −∇j∇ivkp − ∇j∇kvip + ∇j∇pvik 1 = gℓp (∇ ∇ v − ∇ ∇ v − ∇ ∇ v + ∇ ∇ v ) (17) 2 i k jp i p jk j k ip j p ik 1 − gℓp (R v + R v ) . 2 ijkq qp ijpq kq

∂ Exercise 11 Show that if ∂s gij = vij, then:

1. ( ) ∂ 1 R = − ∆ v + ∇ ∇ V − ∇ (div v) − ∇ (div v) , ∂s ij 2 L ij i j i j j i where the Lichnerowicz Laplacian is defined by

∆Lvij + ∆vij + 2Rkijℓvkℓ − Rikvjk − Rjkvik. (18)

3 2. ∂ R = ∇ ∇ v − ∆V − v · R . ∂s ℓ i iℓ 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 |Rc|2 . ∂t 2. ∂ R = ∆ R = ∆R + 2R R − 2R R . ∂t ij L ij ij kijℓ kℓ ik jk 3. ∂ R = ∆R + 2 (B − B + B − B ) (19) ∂t ijkℓ ijkℓ ijkℓ ijℓk ikjℓ iℓjk − (RipRpjkℓ + RjpRipkℓ + RkpRijpℓ + RℓpRijkp) ,

where pr qs Bijkℓ + −g g RipjqRkrℓs = −RpijqRqℓkp. (20)

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