Index

1/4-pinching, 6, 14, 100, 116, 117, 154

Berger’s inequality, 5, 116, 153

Bianchi identities, 2, 3

Bochner formula, 41, 163

Cigar soliton, 16

Convergence theorems for Ricci flow

on S2, 47

on manifolds with curvature in the

interior of

˜,

C 119

on manifolds with strictly 1/4-pinched

curvature, 117

on three-manifolds with positive Ricci

curvature, 70

Covariant derivative, 1, 26

Curvature operator on two-forms, 3

Curvature pinching

global, 4

pointwise, 5

Curvature tensor

algebraic, 53

of a Riemannian manifold, 1

Diameter Sphere Theorem, 8

Differentiable Sphere Theorem, 14, 116,

117, 119

Einstein manifolds

definition, 15

with nonnegative isotropic curvature, 148

Entropy functional on

S2,

40, 41

Evolution equation

of the Levi-Civita connection, 21

of the Ricci tensor, 28

of the Riemann curvature tensor, 27

of the scalar curvature, 29

Exotic sphere, 13

Goldberg-Kobayashi theorem, 137

Hamilton ODE, 54

Hamilton’s convergence criterion, 65

Hamilton’s maximum principle, 56

Hamilton-Ivey estimate, 72

Harmonic map

from S2 into a Riemannian manifold, 9

Laplacian, 17

Holonomy group of a Riemannian manifold

Berger’s classification theorem, 122

definition, 121

Invariant sets for ODE, 51

K¨ ahler manifold, 135

Kulkarni-Nomizu product, 101

Lie derivative, 15

Locally irreducible manifold, 122

Locally symmetric manifold, 122

Nonnegative curvature operator, 3, 100, 165

Nonnegative isotropic curvature, 12, 74, 90,

93, 96, 100, 132, 149

Normal cone to a convex set, 49

Orthonormal frame bundle, 126

Pinching set

definition, 58

examples, 70, 115, 118

175