- CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 423
2.1.4. The infinitesimal structure of locally convex subsets.
The following lemma indicates the infinitesimal convexity of locally con-
vex sets.LEMMA H.18. Let C C M be a locally convex set, let p E C, and let
Vi, 112 E TpM be two unit vectors.(i) Suppose there exists co > 0 such that
expP (sVi) E int (C)
for 0 < s ~ co and i = 1, 2. Then there exists c1 > 0, dependingon M, C, Vi, and 112, such that
expP (s ((1 - t) Vi+ tV2)) E int (C)
for 0 < s ~ c1 and t E [O, 1].(ii) Suppose there exists co > 0 such that
expP (sVi) EC and expP (s112) E int (C)for 0 < s ~co. Given any t E (0, 1], there exists c2 > 0, depending
on M, C, Vi, 112, and t, such that
expP (s ((1 - t) Vi+ t112)) E int (C)
for 0 < s ~ c2.
PROOF. (i) Let U be an open neighborhood of p such that C n U is
convex. Without loss of generality, we may assume that
inj(p) -co< -
2- and B (p, co) c U.
From expP (co Vi) E int (C), for i = 1, 2, and from B (p, co) CU, there exists
an open neighbor hood Ui c U of expP (co Vi) such thatUiCUicint(C).
It follows from the convexity of C n U that for i = 1, 2 there exists an open
neighborhood wi of Vi in the unit sphere(H.6) s;-^1 =i= { w E TpM : 1w1g(p) = 1}
such that
expP (coW) E Ui and expP (sW) EU n int (C)
for all W E Wi and s E (0, co].
Let i: (TpM, g (p)) --t lEn be an isometry and let VilE and wr be the im-
ages in lEn of Vi and Wi under i, respectively. We consider the pointed spaceblow-ups (U,p, A.-^2 g), whose limit as A. --to+ is (TpM, 0, g (p)) = (lEn, o).
Under the blow-up, the limit of the set { expP (sW) : WE Wi ands E (0, col}
is the open cone