1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

364 7. THE REDUCED DISTANCE

Lipschitz functions and other aspects of real analysis on IRn· is the book by

Evans and Gariepy [139]. Many of the results in their book easily extend

to Riemannian manifolds; when this is the case, we state the extensions

without proof. In this section w~ shall assume that (M.n, g) is a complete

Riemannian manifold.

9.1. Locally Lipschitz functions. Recall the definition of differentia-

bility on Riemannian manifolds.

DEFINITION 7.108 (Differentiable function). A function f : M _,IR is

differentiable at p E M if there exists a linear map

Lp: TpM _,IR

such that

lim If (expP (X)) - f (p) - Lp (X)I = O.

X->0 IXI

When this is the case, by definition we write

If (expP (X)) - f (p) - Lp (X)I = o (IXI) as X _, 0.

This implies for every X E TpM that the directional derivative

Dxf =:= lim f (expP (sX)) - f (p) = Lp (X)

s-+O s

exists.

REMARK 7.109. Note that differentiability can be defined more generally

for differentiable manifolds, but in the definition above we chose to endow

the manifold with a Riemannian metric.

Now we list three results in Evans and Gariepy's book. The first is

Theorem 2 in §3.1.2 on p. 81 of [139].

LEMMA 7.110 (Rademacher's Theorem). Let (M.,g) be a Riemannian

manifold. If f : M _, IR is a locally Lipschitz function, then f is differen-

tiable almost everywhere with respect to the Riemannian (Lebesgue) measure.

Secondly, Theorem 5 in §4.2.3 on p. 131 of [139].

LEMMA 7.111 (Locally Lipschitz is equivalent to being in Wi~';i). Let

U be an open set in a Riemannian manifold ( M, g). Then f :. U _, IR is

locally Lipschitz if and only if f E W 1 ~';i (U).

Thirdly, Theorem 2 in §2.4.2 on p. 76 of [139].