1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 419

Since S is convex, we have

/(, ~ { expp (sW) : WE U, s E (0, ll} c S.

Since K is open, we actually have JC c int ( S). Hence U c TpS and we

conclude that TpS is open. D

Note that if Sis not convex, then TpS need not be open. For example,

consider

S ~ { (x,y): y '?:. vr;T} c IP&^2

and p = (0, 0). Then

TpS = { ( o, y) : y > o} ,

which is not open in IP&^2.

2.1.2. Definition of locally convex sets on Riemannian manifolds.

Recall that a subset .E of a Riemannian manifold (Mn, g) is said to be

convex if, for all x, y E .E, every minimal geodesic 'Y in M joining x and y

is contained in .E.

For example:

(1) a subset K C IP&n is convex if and only if for all x, y E K, the line

segment {(1 - t) x + ty: t E [O, 1]} is contained in K,

(2) the set { (x^1 , ... , xn+l) E 5n (1) C JP&n+l : xn+l '?: c }, where c '?: 0,

is convex in the unit sphere sn (1).

Note that the intersection of convex sets is convex.

A subset C C Mn is said to be locally convex if for all x E C there

exists an open neighborhood U of x such that C n U is convex.

For example, the following sets are locally convex but not convex:
(1) the disjoint union of two round balls in IP&n,
(2) a ball of radius r E (1/4, 1/2) in the unit torus 7n ~ IP&n/zn.
Note that a convex set is necessarily connected but a locally convex set
may not be connected.
Since sufficiently small balls are convex in a Riemannian manifold, an
equivalent definition for C to be locally convex is that for all x E C there
exists E > 0 depending on x such that C n B ( x, E) is convex.
Since the notion of interior tangent cone is local, Lemma H.13 implies
that if a set is locally convex, then an interior tangent cone is open.

REMARK H.14. For some other variants on the definition of convexity,
see subsection 1.2 in the next appendix.

2.1.3. Existence of minimal geodesics in locally convex sets.
If C is a connected closed locally convex subset of a Riemannian manifold
(Mn,g), then C is a topological manifold with boundary &C (see pp. 417-
419, including Theorem 1.6, of Cheeger and Gromoll [32]).