1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

462 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

PROPOSITION I.16 (Sublevel set of bp is compact and totally convex). Let

(Mn, g) be a complete noncom pact Riemannian manifold with nonnegative

sectional curvature and let p E M. For any s E [O, oo), the set Cs (p) is

compact and totally convex.

We also recall the following (see Proposition 8.5 of [30] and compare

with Lemma I.8).

PROPOSITION I.17 (Sublevel sets of bp are parallel). Let (Mn,g) be a

complete noncompact Riemannian manifold with nonnegative sectional cur-

vature and let p EM. For any s < t,

(I.4) Cs (p) = { x E Ct : d ( x, 8Ct (p)) 2: t - s}.

In particular, p E 8Co (p) and

8Cs (p) = {x E Ct: d(x,8Ct(P)) = t-s}.

1.3. Splitting and retraction theorems.

There are a number of beautiful theorems concerning the geometry and

topology of complete noncompact manifolds with nonnegative sectional cur-

vature; we now summarize a few of them. We have already stated the

Toponogov comparison theorem in §2 of Appendix G (see Theorem G.33).

In addition, we have the following.

1.3.1. Cheeger-Gromoll splitting theorem.

We have the following fundamental splitting theorem of Cheeger and

Gromoll (see also Theorem 1.162 of [45]), which employs the fact that, when

Re 2: 0, Busemann functions are subharmonic in the sense of distributions

(see Proposition I.12(ii)).

THEOREM I.18 (Cheeger-Gromoll splitting). Suppose (Mn, g) is a com-

plete Riemannian manifold with Re 2: 0 and suppose there is a geodesic line

in Mn. Then (Mn, g) is isometric to ffi.x (Nn-i, h) with the product metric,

where (Nn-i, h) is a Riemannian manifold with Re 2: 0.

REMARK I.19. For the case of Aleksandrov spaces, see Theorem G.45,

which assumes the analogue of nonnegative sectional curvature.

1.3.2. Soul theorem and conjecture.

We recall some basic definitions.

DEFINITION I.20. Let (Nk, h) be a Riemannian manifold and let :E c N
be a submanifold.

(i) We say that :Eis totally geodesic if its second fundamental form
is zero.
(ii) The normal bundle of :Eis
v (:E) ~ {V E TpN : p E :E and (V, W) = 0 for all W E Tp:E}.
For example, a slice Mi x {q} c Mix M2 in a Riemannian product is
totally geodesic.