1547671870-The_Ricci_Flow__Chow

SOME RESULTS IN LOCAL GEOMETRY 289 obtained by taking a~ (Y, a/or) and Z ~ Y -a (8/ar). Then (Z, a/or)= 0 and 8 (Y, \i' f) = Y ...

290 B. SOME RESULTS IN COMPARISON GEOMETRY is singular. In this case, we say q is conjugate top along the geodesic IV defined by ...

SOME RESULTS IN LOCAL GEOMETRY 291 is injective for all p E V'. Now observe that for all sufficiently large i, we have Pi EV', ...

292 B. SOME RESULTS IN COMPARISON GEOMETRY in the limit geodesic tube, which has constant curvature 1, is a geodesic of length 2 ...

l. SOME RESULTS IN LOCAL GEOMETRY 293 where a, b E IR, and Jo is a Jacobi field such that (Jo, T) = O. PROOF. Since 'Y is a geod ...

294 B. SOME RESULTS IN COMPARISON GEOMETRY 1.5. The Rauch comparison theorem. If one has suitable bounds on the sectional curvat ...

LOCAL VERSUS GLOBAL GEOMETRY 295 COROLLARY B.22. Let (Mn,g) be a complete Riemannian manifold with sectional curvatures bounde ...

296 B. SOME RESULTS IN COMPARISON GEOMETRY DEFINITION B.24. If 1 : [0, oo) ----+ Mn is a geodesic with 1 (0) = p, we call 1 ( t ...

LOCAL VERSUS GLOBAL GEOMETRY 297 (1) 0 ~{VE s;;-^1 Mn: p(V) < oo} is an open subset of s;;-^1 Mn; (2) Pio : 0 --> (0, oo ...

298 B. SOME RESULTS IN COMPARISON GEOMETRY further subsequence, we may thus assume Vi' -t V~ #-V. By continuity of the ~xponenti ...

LOCAL VERSUS GLOBAL GEOMETRY 299 2.2. Lifting the metric by the exponential map. Let re =re (p) be the conjugate radius of p E ...

300 B. SOME RESULTS IN COMPARISON GEOMETRY COROLLARY B.36. If (Mn,g) is a complete Riemannian manifold with sectional curvatures ...

LOCAL VERSUS GLOBAL GEOMETRY 301 Assume there is B E IR such that Re 2:: (n - 1) Bg. Wherever r is a smooth generalized distan ...

302 B. SOME RESULTS IN COMPARISON GEOMETRY is continuous in a neighborhood U ~ Mn\ Cut (p) of p and is smooth on U\ {p}. Let "(: ...

BUSEMANN FUNCTIONS 303 of constant sectional curvature H with the same side lengths (a, b, c) such that o: 2 a~ LrpiJ. f3 2 jJ ...

304 B. SOME RESULTS IN COMPARISON GEOMETRY LEMMA B.44. The functions b"f,s are uniformly bounded: for all x E Mn ands 2: 0, lb"f ...

BUSEMANN FUNCTIONS 305 r : [O, oo) -+ JRn is a ray with r (0) = 0 and d1/dt (0) = V, then the associated Busemann function is ...

306 B. SOME RESULTS IN COMPARISON GEOMETRY PROOF. If not, there exists E > 0, a sequence of points Pi E Mn with d (pi, 0) / o ...

BUSEMANN FUNCTIONS 307 3.2. Constructing totally convex half spaces. An important use of Busemann functions is to define arbit ...

308 B. SOME RESULTS IN COMPARISON GEOMETRY In fact, Ts E (0, £), because a (0), a(£) E JH['Y. Thus we can apply the first variat ...