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
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
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