We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.
We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature w
hich substantially improves an earlier estimate in cite{Au2} and give an application on the algebraic entropy of its fundamental group. We also extend the almost minimal volume rigidity of cite{BBCG} to integral Ricci curvature.
The conullity of a curvature tensor is the codimension of its kernel. We consider the cases of conullity two in any dimension and conullity three in dimension four. We show that these conditions are compatible with non-negative sectional curvature on
ly if either the manifold is diffeomorphic to $mathbb{R}^n$ or the universal cover is an isometric product with a Euclidean factor. Moreover, we show that finite volume manifolds with conullity 3 are locally products.
In its most general form, the recognition problem in Riemannian geometry asks for the identification of an unknown Riemannian manifold via measurements of metric invariants on the manifold. We introduce a new infinite sequence of invariants, the firs
t term of which is the usual diameter, and illustrate the role of these global shape invariants in a number of recognition problems.
We study harmonic maps from a 3-manifold with boundary to $mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $pi / 2$. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds.
In this paper we analyze the behavior of the distance function under Ricci flows whose scalar curvature is uniformly bounded. We will show that on small time-intervals the distance function is $frac12$-Holder continuous in a uniform sense. This impli
es that the distance function can be extended continuously up to the singular time.