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Martin Sleziak
Martin Sleziak
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My favorite combinatorial definition for curvature in an $n$-manifold $M$ (given as an abstract simplicial complex) is the angle defect around codimension-2 simplicies, measured using the PL-metric in which all edges have unit length. This makes the curvature at an $(n-2)$-simplex $\sigma$ depend only on the number of $n$-simplices in the manifold with $\sigma$ as a face (the degree of $\sigma$.) That is, $curv(\sigma) = 2\pi - \theta_n deg(\sigma)$ where $\theta_n = \cos^{-1}(1/n)$ is the dihedral angle in a regular $n$-simplex and $deg(\sigma)$ is the degree of $\sigma$.

Using angle defect at codimension-2 simplices goes back to Regge's work, T. Regge (1961). "General relativity without coordinates". Nuovo Cim. 19 (3): 558–571.

In the paper http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r49Positively Curved Combinatorial 3-Manifolds , I prove a Bonnet-Myers type theorem which gives sharp bounds on the edge-diameter of the 1-skeleton of $M$ under the assumption that $curv(\sigma)>0$ at every $(n-2)$-simplex $\sigma$ in $M$. Additionally, I prove a corresponding rigidity result (analogous to the rigid sphere theorems of Toponogov and Cheng) which shows that if such a manifold has the maximum allowed edge-diameter then it must be an $n$-sphere whose triangulation is (almost) completely fixed.

My favorite combinatorial definition for curvature in an $n$-manifold $M$ (given as an abstract simplicial complex) is the angle defect around codimension-2 simplicies, measured using the PL-metric in which all edges have unit length. This makes the curvature at an $(n-2)$-simplex $\sigma$ depend only on the number of $n$-simplices in the manifold with $\sigma$ as a face (the degree of $\sigma$.) That is, $curv(\sigma) = 2\pi - \theta_n deg(\sigma)$ where $\theta_n = \cos^{-1}(1/n)$ is the dihedral angle in a regular $n$-simplex and $deg(\sigma)$ is the degree of $\sigma$.

Using angle defect at codimension-2 simplices goes back to Regge's work, T. Regge (1961). "General relativity without coordinates". Nuovo Cim. 19 (3): 558–571.

In http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r49 , I prove a Bonnet-Myers type theorem which gives sharp bounds on the edge-diameter of the 1-skeleton of $M$ under the assumption that $curv(\sigma)>0$ at every $(n-2)$-simplex $\sigma$ in $M$. Additionally, I prove a corresponding rigidity result (analogous to the rigid sphere theorems of Toponogov and Cheng) which shows that if such a manifold has the maximum allowed edge-diameter then it must be an $n$-sphere whose triangulation is (almost) completely fixed.

My favorite combinatorial definition for curvature in an $n$-manifold $M$ (given as an abstract simplicial complex) is the angle defect around codimension-2 simplicies, measured using the PL-metric in which all edges have unit length. This makes the curvature at an $(n-2)$-simplex $\sigma$ depend only on the number of $n$-simplices in the manifold with $\sigma$ as a face (the degree of $\sigma$.) That is, $curv(\sigma) = 2\pi - \theta_n deg(\sigma)$ where $\theta_n = \cos^{-1}(1/n)$ is the dihedral angle in a regular $n$-simplex and $deg(\sigma)$ is the degree of $\sigma$.

Using angle defect at codimension-2 simplices goes back to Regge's work, T. Regge (1961). "General relativity without coordinates". Nuovo Cim. 19 (3): 558–571.

In the paper Positively Curved Combinatorial 3-Manifolds , I prove a Bonnet-Myers type theorem which gives sharp bounds on the edge-diameter of the 1-skeleton of $M$ under the assumption that $curv(\sigma)>0$ at every $(n-2)$-simplex $\sigma$ in $M$. Additionally, I prove a corresponding rigidity result (analogous to the rigid sphere theorems of Toponogov and Cheng) which shows that if such a manifold has the maximum allowed edge-diameter then it must be an $n$-sphere whose triangulation is (almost) completely fixed.

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Aaron Trout
Aaron Trout
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My favorite combinatorial definition for curvature in an $n$-manifold $M$ (given as an abstract simplicial complex) is the angle defect around codimension-2 simplicies, measured using the PL-metric in which all edges have unit length. This makes the curvature at an $(n-2)$-simplex $\sigma$ depend only on the number of $n$-simplices in the manifold with $\sigma$ as a face (the degree of $\sigma$.) That is, $curv(\sigma) = 2\pi - \theta_n deg(\sigma)$ where $\theta_n = \cos^{-1}(1/n)$ is the dihedral angle in a regular $n$-simplex and $deg(\sigma)$ is the degree of $\sigma$.

Using angle defect at codimension-2 simplices goes back to Regge's work, T. Regge (1961). "General relativity without coordinates". Nuovo Cim. 19 (3): 558–571.

In http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r49 , I prove a Bonnet-Myers type theorem which gives sharp bounds on the edge-diameter of the 1-skeleton of $M$ under the assumption that $curv(\sigma)>0$ at every $(n-2)$-simplex $\sigma$ in $M$. Additionally, I prove a corresponding rigidity result (analogous to the rigid sphere theorems of Toponogov and Cheng) which shows that if such a manifold has the maximum allowed edge-diameter then it must be an $n$-sphere whose triangulation is (almost) completely fixed.