On Combinatorial Riemannian Manifolds

A combinatorial Riemannian manifold (CRM) is a mixture of an Euclidean simplicial complex and a combinatorial manifold. It is an abstract simplicial complex with given edge lengths and flattenings at the vertices, satisfying certain realizability conditions. The describing data are of finite combinatorial and algebraic nature. Some notions of smooth differential geometry, such as the exponential map and the sectional curvature, can be translated to corresponding combinatorial notions. Moreover, there is a canonical PL Riemannian metric on |M| and a CRM with sufficiently fine triangulation is smooth.