scholarly journals Lower curvature bounds, Toponogov's theorem, and bounded topology. II

1987 ◽  
Vol 20 (3) ◽  
pp. 475-502 ◽  
Author(s):  
U. Abresch
Keyword(s):  
Curvature Bounds ◽  
Toponogov’S Theorem

scholarly journals Remarks on Manifolds with Two-Sided Curvature Bounds

2021 ◽  
Vol 9 (1) ◽  
pp. 53-64
Author(s):  
Vitali Kapovitch ◽  
Alexander Lytchak
Keyword(s):  
Distance Functions ◽  
Cut Locus ◽  
Bounded Curvature ◽  
Curvature Bounds ◽  
Positive Reach

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.

scholarly journals Smoothing Riemannian metrics with Ricci curvature bounds

1996 ◽  
Vol 90 (1) ◽  
pp. 49-61 ◽  
Author(s):  
Xianzhe Dai ◽  
Guofang Wei ◽  
Rugang Ye
Keyword(s):  
Ricci Curvature ◽  
Riemannian Metrics ◽  
Curvature Bounds

scholarly journals A∞-weights and compactness of conformal metrics under Ln∕2 curvature bounds

2021 ◽  
Vol 14 (7) ◽  
pp. 2163-2205
Author(s):  
Clara L. Aldana ◽  
Gilles Carron ◽  
Samuel Tapie
Keyword(s):  
Conformal Metrics ◽  
Curvature Bounds

The displacement function and the return map

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Displacement Function ◽  
Semisimple Element ◽  
Fixed Point Set ◽  
Critical Set ◽  
Quantitative Estimates ◽  
Point Set ◽  
Toponogov’S Theorem

This chapter studies the displacement function dᵧ on X that is associated with a semisimple element γ‎ ∈ G. If φ‎″, t ∈ R denotes the geodesic flow on the total space X of the tangent bundle of X, the critical set X(γ‎) ⊂ X of dᵧ can be easily related to the fixed point set Fᵧ ⊂ X of the symplectic transformation γ‎⁻¹φ‎₁ of X. The chapter studies the nondegeneracy of γ‎⁻¹φ‎₁ − 1 along Fᵧ. More fundamentally, this chapter gives important quantitative estimates on how much φ‎ ½ differs from φ‎ ˗½γ‎ away from Fᵧ. These quantitative estimates are based on Toponogov's theorem.

scholarly journals Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

2019 ◽  
Vol 178 (2) ◽  
pp. 319-378 ◽  
Author(s):  
Eric A. Carlen ◽  
Jan Maas
Keyword(s):  
Spectral Gap ◽  
Optimal Transport ◽  
Quantum Systems ◽  
Sobolev Inequalities ◽  
Unified Framework ◽  
Logarithmic Sobolev Inequalities ◽  
Curvature Bounds ◽  
Finite Dimensional ◽  
Dissipative Quantum Systems ◽  
Entropy Inequalities

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.

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