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In this note, we prove that the space of all admissible piecewise linear metrics parameterized by the square of length on a triangulated manifold is a convex cone. We further study Regge’s Einstein-Hilbert action and give a more reasonable definition of discrete Einstein metric than the former version. Finally, we introduce a discrete Ricci flow for three dimensional triangulated manifolds, which is closely related to the existence of discrete Einstein metrics.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v52n2.19.03}, url = {http://global-sci.org/intro/article_detail/jms/13156.html} }In this note, we prove that the space of all admissible piecewise linear metrics parameterized by the square of length on a triangulated manifold is a convex cone. We further study Regge’s Einstein-Hilbert action and give a more reasonable definition of discrete Einstein metric than the former version. Finally, we introduce a discrete Ricci flow for three dimensional triangulated manifolds, which is closely related to the existence of discrete Einstein metrics.