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Riemann Curvature and Ricci Curvature [PDF]

, 2012
Curvatures are the central concept in geometry. The notion of curvature introduced by B. Riemann faithfully reveals the local geometric properties of a Riemann metric. This curvature is called the Riemann curvature in Riemannian geometry. The Riemann curvature can be extended to Finsler metrics as well as the sectional curvature.
Xinyue Cheng, Zhongmin Shen
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Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

Inventiones Mathematicae, 2018
In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts ...
Virginia Agostiniani, Mattia Fogagnolo, L. Mazzieri   +2 more
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Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below

Geometric and Functional Analysis, 2018
We show that non-collapsed Gromov–Hausdorff limits of polarized Kähler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic ...
Gang Liu, G'abor Sz'ekelyhidi
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On 3-manifolds with pointwise pinched nonnegative Ricci curvature

Mathematische Annalen, 2019
There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature
J. Lott
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On orthogonal Ricci curvature

Advances in Complex Geometry, 2019
In this paper we discuss some recent progresses in the study of compact Kähler manifolds with positive orthogonal Ricci curvature, a curvature condition defined as the difference between Ricci curvature and holomorphic sectional curvature.
Lei Ni, F. Zheng
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Ricci curvature and measures [PDF]

Japanese Journal of Mathematics, 2009
In the last thirty years three a priori very different fields of mathematics, optimal transport theory, Riemannian geometry and probability theory, have come together in a remarkable way, leading to a very substantial improvement of our understanding of what may look like a very specific question, namely the analysis of spaces whose Ricci curvature ...
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Ricci Curvature Comparison

1998
In this chapter we shall introduce some of the fundamental theorems for manifolds with lower Ricci curvature bounds. Two important techniques will be developed: relative volume comparison and weak upper bounds for the Laplacian of distance functions.
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The Ricci Curvature

1998
In this chapter we deal with problems concerning Ricci Curvature mainly: Prescribing the Ricci curvature Ricci curvature with a given sign Existence of Einstein metrics.
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Forman's Ricci curvature - From networks to hypernetworks

International Workshop on Complex Networks & Their Applications, 2018
Networks and their higher order generalizations, such as hypernetworks or multiplex networks are ever more popular models in the applied sciences. However, methods developed for the study of their structural properties go little beyond the common name ...
Emil Saucan, Melanie Weber
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Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

Inventiones Mathematicae, 2015
We prove that if (X,d,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(
Fabio Cavalletti, Andrea Mondino
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