# Bakry-\'Emery Ricci curvature of doubly warped product of weighted spaces

## TL;DR

This paper extends the concept of doubly warped products to weighted graphs and smooth measure spaces, establishing curvature bounds and analyzing their properties to connect discrete and continuous geometric settings.

## Contribution

It introduces a new notion of doubly warped products for weighted graphs and smooth measure spaces, and derives curvature bounds in these settings.

## Key findings

- Discrete Bakry-Émery Ricci curvature bounds established
- Examples and applications with toy models provided
- Analysis of curvature saturation at vertices included

## Abstract

We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry-\'Emery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the $\left(R_1,R_2\right)$-doubly warped products of smooth measure spaces and establish $\N$-Bakry-\'Emery Ricci curvature (lower) bounds thereof in terms of those of the factors. At the end of these notes, we present examples and demonstrate applications of warped products with some toy models.