1 LetM be a complete Riemannian manifold of dimensionn. Prove that the following assertions are equivalent:
a. M has constant sectional curvature.
b. M is homogeneous, and its isotropy group at any point is isomorphic toO(n).
c. Given two triples (p, q, r), (p′, q′, r′) of points in M such that d(p, q) = d(p′, q′), d(q, r) = d(q′, r′), d(r, p) = d(r′, p′), there exists an isometry of M that maps the first triple to the second one (Riemannian manifolds with this property are called 3-point homogeneous).
2 Prove that an odd-dimensional compact Riemannian manifold of positive sectional curvature is orientable.
3 LetM be a complete Riemannian manifold of nonpositive curvature. Prove that each homotopy class of curves with given endpoints in M contains a unique geodesic.
4 Consider the disk modelDn ofRHn and let ϕbe an isometry ofRHn.
a. Prove that ϕ uniquely extends to a homeomorphism of the closed ball Dn. (Hint: Use exercise 4 of chapter 3.)
b. Prove that ϕis hyperbolic if and only if its extension toDn admits exactly two fixed points and those lie in the boundarySn−1.
c. Prove that ϕ is parabolic if and only if its extension to Dn admits exactly one fixed point and that lies in the boundarySn−1.
5 LetGbe an Abelian subgroup of the fundamental group of a nonflat space formM. Prove that Gis cyclic.
6 An isometry ϕ of a Riemannian manifold M is called a Clifford translation if the associated displacement function x7→d(x, ϕ(x)) is constant. Prove that:
a. The Clifford translations forRn are just the ordinary translations.
b. The only Clifford translation of RHnis the identity transformation.
c. A linear transformationA∈O(n+ 1) is a Clifford trsnaformation ofSn+1 if and only if either A=±I or there is a unimodular complex number λsuch that half the eigenvalues of A are λand the other half are ¯λ.
7 Let M be a Hadamard manifold. Prove that an isometry ϕ of M is a Clifford translation (cf. exercise 6) if and only if the vector field X on M given by expp(Xp) =ϕ(p) is parallel.
8 Extend Preissmann’s theorem 6.5.11 to show that every solvable subgroup of the fundamental group of a compact Riemannian manifold of negative curvature must be infinite cyclic.
9 In this exercise, we prove that a compact homogeneous Riemannian manifold M whose Ricci tensor is negative semidefinite everywhere is isometric to a flat torus.
a. Use exercise 8 of chapter 5 to show that the identity component of the isometry group ofM is Abelian.
b. Check that M can be identified with an n-torus equipped with a left-invariant Riemannian metric.
c. Show that an n-torus equipped with a left-invariant Riemannian metric admits a global parallel orthonormal frame and hence is flat.
10 A Riemannian manifold M is called locally symmetric if every point p∈M admits a normal neighborhood V and an isometry ϕ:V →V such thatϕ(p) =p and dϕp =−id.
a. Show that space forms and Lie groups with bi-invariant metrics are locally symmetric. (Hint:
for the second example, use group inversion.)
b. Prove that the curvature tensor of a locally symmetric manifold is parallel. (Hint: Use the version of equation (4.2.6) for ∇R.)
11 Let M be a Riemannian manifold with curvature tensorR.
a. Prove thatRis parallel if and only if for every smooth curve γ inM and parallel vector fields X,Y,Z,W along γ we have thathR(X, Y)Z, Wiis constant.
b. Prove that ifR is parallel then the Jacobi equation along a geodesic has constant coefficients in a suitable basis.
12 In this exercise, we prove the converse of the result of exercise 10(a).
a. LetM and ˜M be a Riemannian manifolds with parallel curvature tensors. Suppose there are pointsp ∈M, ˜p∈M˜ and a linear isometry f :TpM →Tp˜M˜ such that takes any 2-plane in TpM to a 2-plane inTp˜M˜ with the same sectional curvature. Prove that there exists normal neighborhoods V, ˜V of p, ˜p, resp., and an isometry F : V → V˜ such that F(p) = ˜p and dFp =f. (Hint: combine the idea in the proof of Theorem 6.2.1 with exercise 11(b)).
b. Prove that a Riemannian manifold with parallel curvature tensor is locally symmetric. (Hint:
Apply part (a) to M = ˜M and f =−id.)
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Index
ǫ-totally normal neighborhood, 51 action, 20
isotropy group, 20 orbit, 20
orbit space, 20 proper, 20 smooth, 20 transitive, 22 adjoint representation
of Lie algebra, 19 of Lie group, 19
Allamigeon-Warner manifold, 109 almost complex structure, 89 almost K¨ahler manifold, 89 Bianchi identity
first, 80 second, 84 biquotient, 126 Blaschke
conjecture, 110 manifold, 110 Cartan-Killing form, 90 center of mass, 122 Clifford translation, 127 complex projective space, 34 complex structure, 88 conjugate locus, 102 conjugate point, 102
first, 105 conjugate value, 102 connection, 43
Christoffel symbols, 45
covariant derivative along a curve, 47 Levi-Civit`a, 45
Koszul formula, 45 convex function, 120
strictly, 120 coordinate vector, 3 covering
smooth, 8
topological, 7
covering transformation, 8 curvature
Ricci, 83 scalar, 83 sectional, 81 tensor, 79 cut locus, 72
deck transformation, 8 diameter, 69
diffeomorphism, 2 local, 2
differential of a map, 5 displacement function, 123 divergence, 93
embedding, 6 energy, 95
Euclidean space, 28 exponential map, 50 flat torus, 30
Fubini-Study metric, 34 fundamental group, 7 Gauss
lemma, 63, 103 geodesic, 49
equation, 49
is locally minimizing, 66
local existence and uniqueness, 50 gradient, 93
Green identities, 94 Hadamard manifold, 121 harmonic
function, 94 Heisenberg algebra, 15 Hermitian metric, 89 Hessian, 94
homogeneous space, 22 hyperbolic manifold, 116
immersion, 5 index form, 99 injectivity radius, 71 isometric immersion, 28 isometry group, 27 isotropy group, 20
isotropy representation, 37 Jacobi
equation, 100 field, 100 K¨ahler manifold, 89 Killing form, 90 Killing vector field, 53 Klein bottle, 32 Laplacian, 94 lens space, 116 Lie algebra, 15 Lie bracket, 12 Lie group, 14
exponential map, 16 homomorphism, 17 local section, 22
manifold smooth, 1 map
differential, 5 proper, 6 smooth, 2
normal neighborhood, 51 orbit, 20
orbit space, 20
Poincar´e conjecture, 126 real hyperbolic space, 29 real projective space, 32 Ricci
flow, 126
Riemannian covering, 31 Riemannian manifold, 25
as metric space, 65 complete, 69 conformally flat, 29 geodesically complete, 67 homogeneous, 36
isotropic, 60
normal homogeneous, 37 submanifold, 28
Riemannian measure, 93
Riemannian metric, 25 bi-invariant, 35 conformal, 29 existence, 27 flat, 28
homothetic, 29 induced, 28 left-invariant, 35 product, 29, 55 pulled-back, 28 right-invariant, 35 Riemannian submersion, 33 Schur lemma, 82
smooth manifold, 1 homogeneous, 22 space form, 113 sphere, 29 submanifold
embedded, 2 immersed, 5 submersion, 6 tangent bundle, 5 tangent space, 3 Teichm¨uller space, 117 tensor
curvature, 79 Ricci, 82 theorem of
Bieberbach, 115 Bonnet-Myers, 119 Cartan, 122 divergence, 94
Hadamard-Cartan, 120 Hopf-Rinow, 67 inverse function, 5 Jacobi-Darboux, 104 Killing-Hopf, 115 Myers-Steenrod, 27 Preissmann, 123 Synge, 118
totally normal neighborhood,seeǫ-totally normal neigh-borhood
variation of curve, 96
first variation of energy, 97 second variation of energy, 99 variational vector field, 97 vector field
f-related, 13 flow, 11
incompressible, 94 integral curve, 10 Lie bracket, 12