Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature
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A combinatorial analogue of Bochner's theorems is derived, which demonstrates that there are topological restrictions to a space having a cell decomposition with everywhere positive curvature.Abstract:
. In this paper we present a new notion of curvature for cell complexes. For each p , we define a p th combinatorial curvature function, which assigns a number to each p -cell of the complex. The curvature of a p -cell depends only on the relationships between the cell and its neighbors. In the case that p=1 , the curvature function appears to play the role for cell complexes that Ricci curvature plays for Riemannian manifolds. We begin by deriving a combinatorial analogue of Bochner's theorems, which demonstrate that there are topological restrictions to a space having a cell decomposition with everywhere positive curvature. Much of the rest of this paper is devoted to comparing the properties of the combinatorial Ricci curvature with those of its Riemannian avatar.read more
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References
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Book
Curvature and Betti numbers
健太郎 矢野,Salomon Bochner +1 more
TL;DR: In this paper, the authors proposed a pseudo-harmonic tensors and pseudo-killing tensors in metric Manifolds with Torsion, which can be seen as a kind of semi-simple group spaces.
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A Recurring Theorem on Determinants
TL;DR: In this article, a recurring theorem on determinants is proposed for determining determinants, and it is shown that theorem holds for all determinants in the case of determinants.
Journal ArticleDOI
Vector fields and Ricci curvature
TL;DR: In this article, it was shown that the Ricci curvature of a Riemannian metric is either everywhere positive or everywhere negative, and the divergence and curl both vanish.