• Corpus ID: 119590371

# On the geometry of Riemannian manifolds with density

@article{Wylie2016OnTG,
title={On the geometry of Riemannian manifolds with density},
author={William C. Wylie and Dmytro Yeroshkin},
journal={arXiv: Differential Geometry},
year={2016}
}
• Published 25 February 2016
• Mathematics
• arXiv: Differential Geometry
We introduce a new geometric approach to a manifold equipped with a smooth density function that takes a torsion-free affine connection, as opposed to a weighted measure or Laplacian, as the fundamental object of study. The connection motivates new versions of the volume and Laplacian comparison theorems that are valid for the 1-Bakry-Emery Ricci tensor, a weaker assumption than has previously been considered in the literature. As applications we prove new generalizations of Myers' theorem and…

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## References

SHOWING 1-10 OF 45 REFERENCES

### Riemannian Geometry

THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss's

### CODAZZI TENSOR FIELDS, CURVATURE AND PONTRYAGIN FORMS

• Mathematics
• 1983
for arbitrary vector fields X, Y, Z. In this case, the self-adjoint section B of End TM, characterized by g(BX, Y) = b(X, Y), will also be called a Codazzi tensor. The Codazzi tensor b will be called

### Sectional curvature for Riemannian manifolds with density

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove

### Ricci curvature for metric-measure spaces via optimal transport

• Mathematics
• 2004
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the

### Positive weighted sectional curvature

• Mathematics
• 2014
In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We

### Poincaré and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary

• Mathematics
• 2014
It is well known that by dualizing the Bochner{Lichnerowicz{Weitzenbock for- mula, one obtains Poincar e-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry{

### Classification of irreducible holonomies of torsion-free affine connections

• Mathematics
• 1999
The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified, thus completing the work on this subject

### Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary

• Mathematics
• 2013
It is known that by dualizing the Bochner–Lichnerowicz–Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry–Émery

### Comparison geometry for the Bakry-Emery Ricci tensor

• Mathematics
• 2007
For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is

### Manifolds with Density and Perelman's Proof of the Poincaré Conjecture

1. DEFINITIONS. A manifold with density is a Riemannian manifold Mn (n dimensional surface with an infinitesimal arclength ds that you can use to compute lengths, areas, and volumes) with a positive