1547671870-The_Ricci_Flow__Chow

APPENDIX B

Some results In comparison geometry

We begin this appendix by recalling basic results in comparison geometry
that are useful in the study of limits of dilations about a singularity of the
Ricci flow. Because these are standard results, we shall in some cases omit or
merely sketch the proofs. We assume throughout that (Mn, g) is a complete
Riemannian manifold.

1. Some results in local geometry

We begin with a collection of results that hold at small length scales,
where a complete Riemannian manifold (Mn, g) is ' almost Euclidean'.

1.1. The Gauss Lemma and radial vector fields. Recall that for

each point p E Mn and tangent vector V E TpMn, there is a unique

constant-speed geodesic

IV : [O, 00) ---t Mn

emanating from p such that IV (0) = p and 'Yv (0) = V. Given p E Mn, let

expp : TpMn ---+ Mn denote the exponential map defined by

expp (V) ~IV (1).

By the uniqueness of geodesics with given initial data, one has

IV (t) = ltV (1),

so that

IV (t) = expp (tV).

Observe that (expp)*: T 0 (TpMn)---+ TpMn is the natural isomorphism

between the two vector spaces. (We may in fact regard it as the identity

map.) So by the inverse function theorem, there exists c: > 0 such that

expPIB(6,e): B(O,c:)---+ expP (B(O,c:)) c Mn

is a diffeomorphism, where B(V, r) denotes the open ball with center V E

TpMn and radius r. In fact, even more is true; expP preserves orthogonality

to radial directions, and hence the length of radial vectors.

LEMMA B.l (Gauss). If p E Mn and V, W E TpMn are such that

(V, W) = 0, then

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