Abstract
We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.
AbstractWe study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $$C^*$$
C
∗
-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.
Rigidity of Einstein Metrics as Critical Points of Some Quadratic Curvature Functionals on Complete Manifolds
