Abstract
The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Regges Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein-Hilbert-Regge functional on the space of metrics and on discrete conformal classes of metrics.
Original language | English (US) |
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Pages (from-to) | 108-124 |
Number of pages | 17 |
Journal | Differential Geometry and its Application |
Volume | 29 |
Issue number | 1 |
DOIs | |
State | Published - Feb 2011 |
Keywords
- Constant scalar curvature
- Einstein
- Einstein-Hilbert
- Piecewise flat
- Regge
- Yamabe
ASJC Scopus subject areas
- Analysis
- Geometry and Topology
- Computational Theory and Mathematics