1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. SHARAFUTDINOV RETRACTION THEOREM 463

DEFINITION l.21 (Soul). A submanifold S of a complete noncompact

manifold (Mn, g) is called a soul if it is a compact, totally convex, totally

geodesic submanifold.

Since Sis compact, we have dim (S) <dim (M).

The following result, known as the soul theorem, is fundamental in the

understanding of manifolds with nonnegative curvature.

THEOREM I.22 (Soul theorem). Let (Mn,g) be a complete noncompact

Riemannian manifold ..

(1) (Cheeger and Gromoll [32], Theorem 1.11 and Theorem 2.2) If

the sectional curvature of g is nonnegative, then there exists a soul

S C M and M is diffeomorphic to its normal bundle v (S).

(2) (Sharafutdinov [172]) If sect (g) 2:: 0, then any two souls are iso-

metric.

(3) (Gromoll and Meyer [77]) If the sectional curvature of g is positive,

then any soul is a point and hence Mn is diffeomorphic to ffi.n.

Given p E M, recall that (bp)min ~ minxEM bp (x) > -oo is attained

since bp is a proper function bounded from below. Moreover, the subset
b;;^1 ( (bP)min) contains a soul.
Generalizing Theorem I.22(3), Perelman [148] proved the following,
known as the soul conjecture.

THEOREM I.23 (Soul conjecture). If (Mn,g) is a complete noncompact

Riemannian manifold with nonnegative sectional curvature everywhere and

with positive sectional curvature at some point, then the soul is a point.

1.3.3. Sharafutdinov retraction map.
An important tool in the study of manifolds with nonnegative curvature
is the Sharafutdinov retraction map, which we define below.
Let (Mn, g) be a complete noncom pact Riemannian manifold with non-
negative sectional curvature. Fix a point p E M and let bp be the corre-
sponding Busemann function. By Proposition I.12 and Corollary I.13, the

function -bp is concave and bounded from above on M.

Specializing Theorem H.59 to the Busemann function, we have the fol-
lowing (which is used in the proof of Proposition 18.33).

THEOREM I.24 (Sharafutdinov's retraction for level sets of the Busemann
function). Let (Mn, g) be a complete noncompact manifold with nonnegative

sectional curvature and fix a point p E M. Then for any a2 < ai :S ( -bp )max

there exists a distance-nonincreasing map

between the level sets of the Busemann Junction bp. The same result holds
for the Busemann function b'Y associated to a ray 'Y.