Fathi Zohreh, Lakzian Sajjad
Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran.
Department of Mathematical Sciences, Isfahan University of Technology (IUT), Isfahan, 8415683111 Iran.
J Geom Anal. 2022;32(3):79. doi: 10.1007/s12220-021-00745-7. Epub 2022 Jan 12.
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-Émery 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 -doubly warped products of smooth measure spaces and establish -Bakry-Émery 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.
我们引入了加权图的双扭曲积的概念，它与黎曼几何背景下的双扭曲积是一致的。我们根据组成图的曲率，为这种扭曲积建立了各种离散的巴克利 - 埃默里里奇曲率 - 维数界。这需要对所涉及的二次型进行细致分析，从而促使我们引入一些关键概念，比如顶点处的曲率饱和度。本着全面性以及提供一个参考框架的精神，我们还引入了光滑测度空间的(\infty -)双扭曲积，并根据因子的(\infty -)巴克利 - 埃默里里奇曲率（下界）来建立其(\infty -)巴克利 - 埃默里里奇曲率（下界）。在这些笔记的结尾，我们给出示例，并通过一些简单模型展示扭曲积的应用。
