Abstract
We prove a theorem that describes all possible tensor-valued natural operations in the presence of a linear connection and an orientation in terms of certain linear representations of the special linear group. As an application of this result, we prove a characterization of the torsion and curvature operators as the only natural operators that satisfy the Bianchi identities.
Highlights
We focus our attention on the vector space of tensor-valued natural operations that can be performed in the presence of a linear connection and an orientation
As an example of this philosophy, in the final section, we characterize the torsion and the curvature as the only natural tensors satisfying the Bianchi identities (Corollary 13 and Theorem 15). These results generalize analogous statements that were recently proven in [16], where we studied natural tensors associated with a linear connection
The purpose of this section is twofold: On the one hand, we present the notion of natural operation (Definition 7); our definition strongly differs from the standard one, it is equivalent to it ([18])
Summary
As an example of this philosophy, in the final section, we characterize the torsion and the curvature as the only natural tensors satisfying the Bianchi identities (Corollary 13 and Theorem 15) These results generalize analogous statements that were recently proven in [16], where we studied natural tensors associated with a linear connection. The non-specialist may find it difficult to understand the precise meaning of some statements of this book due to the functorial language and the generality of its setting For this reason, we outlined in [16] the foundations of an alternative approach, which we hope will be accessible to a wider audience.
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