Exercises

1 LetM be ann-dimensional Riemannian manifold of constant curvatureκ. Compute that Ric = (n−1)κg and scal =n(n−1)κ.

2 Let g and ¯g be two Riemannian metrics in the smooth manifold M such that ¯g = λg for a constant λ > 0. Show that the curvature tensor, the sectional curvature, the Ricci tensor and the scalar curvature of the Riemannian manifolds (M,¯g) and (M, g) are related by the following equations:

R=R, K =λ⁻¹K, Ric = Ric and scal =λ⁻¹scal.

3 Use the symmetries of the curvature tensor to show that the Ricci tensor determines the curva-ture tensor in a Riemannian manifold of dimension 3.

4 Let M be a Riemannian manifold with the property that for any two points p, q ∈ M, the parallel transport map fromptoq along a piecewise smooth curveγ joiningptoq does not depend on γ. Prove that M must be flat.

5 As a partial converse to the previous exercise, suppose M is a flat manifold, p,q ∈M, and γ0, γ₁ are two smooth curves joining ptoq. Prove that ifγ₀ and γ₁ are smoothly homotopic with the endpoints fixed, then the parallel transport maps fromp toq along γ₀ and along γ₁ coincide.

6 Prove that the curvature tensor of CPⁿ is

R(X, Y)Z =−hX, ZiY +hY, ZiX+hX, JZiJY − hY, JZiJX+ 2hX, JYiJZ for vector fieldsX,Y,Z on CPⁿ. (Hint: Use formula (4.5.10).)

7 Prove that the curvature tensor and the Ricci tensor of a K¨ahler manifold (M, g, J) satisfy the following identities:

R(X, Y)J =JR(X, Y), R(JX, JY) =R(X, Y) and Ric(JX, JY) = Ric(X, Y), for all vector fields X and Y onM.

8 Prove that the curvature tensor of a Riemannian manifold satisfies the following identities:

a. For tangent vectorsx,y,z and w, we have

6hR(x, y)z, wi = hR(x, y+z)(y+z), wi − hR(x, y−z)(y−z), wi +hR(y, x−z)(x−z), wi − hR(y, x+z)(x+z), wi b. For tangent vectorsa,b,c, we have

4hR(a, b)a, ci=hR(a, b+c)a, b+ci − hR(a, b−c)a, b−ci Deduce an alternative proof of Corollary 4.2.3.

9 (Riemannian volume) Let (M, g) be an oriented Riemannian manifold of dimension n. Let E = (E1, . . . , En) a positively oriented orthonormal frame on an open subsetU (that is,E1, . . . , En

are smooth vector fields defined onU which are orthonormal at each point), and let (θ¹, . . . , θⁿ) be the dual coframe of 1-forms onU. Define the n-formω_E =θ¹∧ · · · ∧ · · ·θⁿ onU.

a. Prove that for another positively oriented orthonormal frameE^′ defined onU^′ we have ω_E = ω_E^′ on U ∩U^′. Deduce that there exists a smooth differential form vol_M of degree n on M such that

(volM)p(e1, . . . , en) = 1

for every positively oriented orthonormal basise₁, . . . , e_n ofT_pM and allp∈M. Then-form vol_M is called thevolume form of (M, g) and the associated measure is called theRiemannian measure on M associated to g.

b. Show that for a positively oriented basisv₁, . . . , v_n ofT_pM, we have (vol_M)_p(v₁, . . . , v_n) =q

det (g_p(v_i, v_j)).

Deduce that, in local coordinates (U, ϕ= (x¹, . . . , xⁿ)), vol_M =q

det(g_ij)dx¹∧ · · · ∧dxⁿ. 10 Let (M, g) be an n-dimensional Riemannian manifold.

a. For any smooth function f : M → R, the gradient of f is the smooth vector field gradf defined byg((gradf)_p, v) =df_p(v) for all v∈T_pM and all p∈M. Prove that

grad(f₁+f₂) = gradf₁+ gradf₂ and grad(f₁f₂) =f₁gradf₂+f₂gradf₁ for all smooth functionsf₁,f₂ onM.

b. For any smooth vector field X on M, the divergence of X is the smooth function divX = trace (v7→ ∇vX). Prove that

div (X+Y) = divX+ divY and div (f X) =hgradf, Xi+fdivX for all smooth fuctionsf and smooth vector fields X,Y on M.

c. For any smooth functionf onM, theLaplacian off is the smooth function ∆f = div gradf.

The function f is called harmonic is ∆f = 0. Prove that

∆(f1f2) =f1∆f2+hgradf1,gradf2i+f2∆f1

for all smooth functionsf₁,f₂ onM.

d. For any smooth function f on M, the Hessian off is the (0,2)-tensor Hess(f) =∇df. Prove that

Hess(f)(X, Y) =X(Y f)−(∇XY)f and

Hess(f)(X, Y) = Hess(f)(Y, X)

for all smooth vector fields X, Y on M. Show also that the trace of the Hessian coincides with the Laplacian.

11 (Divergence theorem) LetM be an oriented Riemannian manifold.

a. Prove that for any smooth vector field

L_X(dV) = (divX)dV

where dV denotes the volume form volM. A vector field is called incompressible if it is divergence free. Deduce that a vector field is incompressible if and only if its local flows are volume preserving.

b. Suppose now Ω is a domain in M with smooth boundary and let ∂Ω be oriented by the outward unit normal ν. Denote the Riemannian volume form of ∂Ω by dS. Use Stokes’

theorem to show that for any compactly supported smooth vector field X on M we have Z

Ω

divX dV = Z

∂ΩhX, νidS

12 (Green identities) LetM be an oriented Riemannian manifold and let Ω be a domain inM as in exercise 11.

a. Prove the “integration by parts formula”

Z

Ω

f₁∆f₂dV + Z

Ωhgradf₁,gradf₂idV = Z

∂Ω

f₁∂f2

∂ν dS

for any compactly supported smooth functions f₁, f₂ on M. Deduce the weak maximum principle: if f is compactly supported and sub- or super-harmonic (i.e. ∆f ≥0 or ∆f ≤0) thenf is constant. (Hint: first show ∆f = 0; then apply integration by parts tof =f1 =f2

and Ω =M.)

b. Prove that Z

Ω

(f1∆f2−f2∆f1)dV = Z

∂Ω

f1∂f2

∂ν −f2∂f1

∂ν

dS

for any compactly supported smooth functionsf₁,f₂ onM. Deduce that iff₁ andf₂ are two eigenfunctions of the Laplacian on a compact oriented Riemannian manifoldM associated to different eigenvaluesλ₁,λ₂, resp., thenf₁andf₂are orthogonal in the sense thatR

Mf₁f₂dV = 0.

C H A P T E R 5

Variational calculus

5.1 Introduction

We continue to study the problem of minimization of geodesics in Riemannian manifolds that was started in chapter 3. We already know that geodesics are the locally minimizing curves. Also, long segments of geodesics need not be minimizing, and the study of this phenomenon in complete Riemannian manifolds motivates the definition of cut locus.

Herein we take a different standpoint in that we consider finite segments of curves. Namely, consider a complete Riemannian manifoldM. Given two pointsp,q∈M, the Hopf-Rinow theorem ensures the existence of at least one minimizing geodesic γ joining p and q. It follows that γ is a global minimum for the length functional L defined in the space of piecewise smooth curves joining p and q. Of course, the calculus approach to finding global minima of a function is to differentiate it, compute critical points and decide which of them are local minima by using the second derivative. In our case, the apparatus of classical calculus of variations can be applied to carry out this program.

To begin with, we show that the critical points of the length functional in the space of piecewise smooth curves joiningpandqare exactly the geodesic segments, up to reparametrization. The main result of this chapter is the Jacobi-Darboux theorem that gives a necessary and sufficient condition for a geodesic segment between p and q to be a local minimum for L. In order to prove this theorem, we introduce Jacobi fields and conjugate points. Finally, we study the relation between the concepts of cut locus and conjugate locus. These results will be generalized in chapter 8,¹ where we will prove the Morse index theorem.

Throughout this chapter, (M, g) denotes a Riemannian manifold.