 
Summary: Math. Ann. 303, 297306 (1995)
t~ SpringerVerlas1995
A comparisonestimate of Toponogov type
for Ricci curvature
Xianzhe Dai 1,*, Guofang Weiz,**
SDepartment of Mathematics, University of SouthernCalifornia,Los Angeles,CA 90089, USA
(email: xdai@math.usc.edu)
2Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
(email: wei@math.ucsb.edu)
Received: 17 May 1994/Revised version: 17 October 1994
Mathematics Subject Classification (1991): 53C20
1 Introduction
It seems a natural question to ask to what extent the results and tools for
sectional curvature remain valid for Ricci curvature. There is rapid progress
in both positive and negative directions. Toponogov Comparison Theorem has
been the most powerful tool in the study of sectional curvature, underlying the
proof of the Soul Theorem, the diameter sphere theorem, the uniform estimate
of betti numbers and the finiteness theorems. Toponogov Comparison Theo
rem is also the characterizing property of lower (or upper) sectional curvature
bounds, which led to generalizations of the concept of (sectional) curvature
