Holonomy of Manifolds with Density

TL;DR

This paper explores the holonomy groups of a specific connection related to the Bakry-Émery Ricci curvature, classifying all in 2D and presenting two infinite families, advancing understanding of geometric structures with density.

Contribution

It classifies holonomy groups in dimension 2 and introduces two infinite families, expanding the knowledge of geometric structures with density.

Findings

Classified all possible holonomy groups in dimension 2.

Identified two infinite families: SL_n(ℝ) and SO^+(p,q).

Provided examples and properties of holonomy groups for the connection.

Abstract

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 curvature, and also Wylie's $\bar{\mathrm{sec}}_f$. In particular we classify all possible holonomy groups in dimension 2 and also provide two infinite families: $SL_n(\mathbb{R})$ and $SO^+(p,q)$.