Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds - Mathematics > Differential GeometryReportar como inadecuado




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Abstract: A piecewise flat manifold is a triangulated manifold given a geometry byspecifying edge lengths lengths of 1-simplices and specifying that allsimplices are Euclidean. We consider the variation of angles of piecewise flatmanifolds as the geometry varies in a particular way, which we call a conformalvariation. This variation generalizes variations within the class of circleswith fixed intersection angles such as circle packings as well as otherformulations of conformal variation of piecewise flat manifolds previouslysuggested. We describe the angle derivatives of the angles in two and threedimensional piecewise flat manifolds, giving rise to formulas for thederivatives of curvatures. The formulas for derivatives of curvature resemblethe formulas for the change of scalar curvature under a conformal variation ofRiemannian metric. They allow us to explicitly describe the variation ofcertain curvature functionals, including Regge-s formulation of theEinstein-Hilbert functional total scalar curvature, and to consider convexityof these functionals. They also allow us to prove rigidity theorems for certainanalogues of constant curvature and Einstein manifolds in the piecewise flatsetting.



Autor: David Glickenstein

Fuente: https://arxiv.org/







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