A comparison-estimate of Toponogov type for Ricci curvature

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 Theorem is also the characterizing property of lower (or upper) sectional curvature bounds, which led to generalizations of the concept of (sectional) curvature bounds to non-smooth space. Thus it could not possibly hold for Ricci curvature, which makes problems a lot harder for Ricci curvature (from a geometric point of view). It also makes the Ricci curvature very different from the sectional curvature, as one gradually comes to realize. It is interesting then that we find a comparison estimate of Toponogov type for Ricci curvature. A hinge in a complete Riemannian manifold consists of two geodesic segments Vi,Y2 such that "yI(II)= ~2(0) 9 We denote it by (~1,~2,~), where = Z(-) ,~( / l ) ,~(O)) . We will be using the following modified version of the conjugate radius function Pc(P):