Smooth Limits of Piecewise-Linear Approximations

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We consider particular types of discrete approximations to tensor fields on manifolds suggested by triangulations. The approximations are objects of finite geometrical extent, parameterized by a finite set of numbers, so they are suitable for numerical computations. We study the limiting behaviour of sequences of approximations and construct the theory so that the limits are tensor fields on the manifold. We propose a Cauchy criterion for our approximations, which guarantees convergence to a limit. The specific examples include geodesic approximation to Riemannian and pseudo-Riemannian manifolds.