Applications of Alexandrov spaces to Riemannian geometry

Question

I am an expert neither in Riemannian geometry nor in Alexandrov spaces. I am wondering what are the applications of Alexandrov spaces to more classical Riemannian geometry.

For example one can show that there exist only finitely many homeomorphisms types of closed smooth Riemannian manifolds of dimension $n$, diameter at most $D$, sectional curvature at least $\kappa$, and volume at least $v>0$ for fixed parameters $n,D,\kappa,v$. The proof uses the Gromov compactness theorem and the Perelman stability theorem.

Answer 1

The two sources of applications come from two sources of examples of Alexandrov spaces:

1. Limits of Riemannian manifolds with lower curvature bound.

2. Quotients of Riemannian manifolds by an isometric group action with closed orbits.

Your example, the finiteness theorem, is of type 1. Another example is the upper bound on integral of scalar curvature of Riemannian manifold in terms of its lower bound on sectional curvature, diameter and dimension, see my paper. There are many more, in fact (1) provides the main sourse of applications so far.

The examples of the second type include the classification of 4-dimesional Riemannian manifolds with positive/non-negative curvature; it was done by Grove and Wilking here and based on earlier result of Hsiang and Kleiner. Yet anther example is the optimal bound for the number of certain type finite subgroups up to conjugacy in a crystallographic group, see this paper by Lebedeva (which is build on an observation of Perelman, which is build on an combigeometrical problem of Erdős, Danzer and Grünbaum).