Defining a simplex on a Riemannian manifold is not straightforward, the convex hull for example is not useful, in all but a few very special cases. In this talk we'll see the construction of Riemannian n-simplices on n-manifolds and conditions for them being homeomorphic to the standard Euclidean n-simplex (which we'll call non-degeneracy conditions). We'll discuss the tools that are used in this definition and non-degeneracy conditions, such as the Toponogov comparison theorem and Riemannian centres of mass.

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