The Number of Tensor Differential Invariants of a Riemannian Metric
نویسنده
چکیده
We determine the number of functionally independent components of tensors involving higher–order derivatives of a Riemannian metric. A central problem in Riemannian geometry is to determine the number of functionally independent invariants which can be constructed out from a Riemannian metric and its derivatives up to a given order. Due to the inmediate applications of these invariants as Lagrangians in the formulation of alternative gravitational theories, the work have been mainly oriented to scalar invariants [1]–[5]. In this work we determine the number of functionally independent components of tensors involving higher–order derivatives of a Riemannian metric. As a starting point we adopt the taxonomic [6] definition of a tensor based on the transformation rule of its components: a tensor is something which transforms like a tensor. This means that in the corresponding transformation rule only the transformation matrix, Xα = ∂x /∂y appears. When considering derivatives of the metric, up to a certain order d, derivatives of the transformation matrix will appear. By a simple counting it is possible to determine the number of relations not involving derivatives of the transformation matrix. This number turns to be the number of components of a tensor, of rank (d + 2), involving derivatives of the metric. Let us now consider in detail the determination of the number of independent components of tensors involving derivatives of the metric. As 1 [email protected]
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