# 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} }

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…

## 18 Citations

### The Weighted Connection and Sectional Curvature for Manifolds With Density

- Mathematics
- 2017

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion-free connection introduced recently by the last two authors. We…

### Laplacian comparison theorem on Riemannian manifolds with modified m-Bakry-Emery Ricci lower bounds for $m\leq1$

- Mathematics
- 2021

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth n-dimensional Riemannian manifold having a lower bound of modified m-Bakry-Émery Ricci…

### The Euler Class from a General Connection, Relative to a Metric

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- 2021

We extend the well-known formula for the Euler class of a real oriented even-dimensional vector bundle in terms of the curvature of a metric connection to the case of a general linear connection…

### An Integral Formula for Affine Connections

- Mathematics
- 2016

In this article, we introduce a 2-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new…

### Curvature-dimension bounds for Lorentzian splitting theorems

- MathematicsJournal of Geometry and Physics
- 2018

### Comparison Theorems on Weighted Finsler Manifolds and Spacetimes with ϵ-Range

- MathematicsAnalysis and Geometry in Metric Spaces
- 2022

Abstract We establish the Bonnet–Myers theorem, Laplacian comparison theorem, and Bishop–Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of…

### Quantitative Estimate of Diameter for Weighted Manifolds Under Integral Curvature Bounds and $$\varepsilon $$-Range

- MathematicsResults in Mathematics
- 2022

In this article, we extend the compactness theorems proved by Sprouse [12] and Hwang–Lee [3] to a weighted manifold under the assumption that the weighted Ricci curvature is bounded below in terms of…

### Holonomy of Manifolds with Density

- Mathematics
- 2020

In this paper we discuss some examples and general properties of holonomy groups of $\nabla^\varphi$ introduced by Wylie and the author, the connection corresponding to the $N=1$ Bakry-\'Emery Ricci…

### Concentration of $1$-Lipschitz functions on manifolds with boundary with Dirichlet boundary condition

- Mathematics
- 2017

In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative $1$-Lipschitz functions vanishing on the…

### The Bakry-Émery Ricci Tensor: Application to Mass Distribution in Space-time

- MathematicsGravitation and Cosmology
- 2021

Abstract The Bakry-Émery Ricci tensor gives an analogue of the Ricci tensor for a Riemannian manifold with a smooth function. This notion motivates a new version of Einstein’s field equation in which…

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