Tensor Generalizations of Affine Symmetry Vectors

TL;DR

This paper introduces affine symmetry tensors as a generalization of affine vectors, providing new identities and clarifying their relation to geodesic deviation and homothetic tensors in differential geometry.

Contribution

It proposes a formal definition for affine symmetry tensors, proves an identity relating their second derivatives to curvature, and explores their inclusion relations and connections to geodesic deviation.

Findings

Derived an identity linking second derivatives of affine symmetry tensors to curvature.

Established inclusion relations among affine, homothetic, and other symmetry tensors.

Clarified the connection between affine symmetry tensors and solutions to geodesic deviation equations.

Abstract

A definition is suggested for affine symmetry tensors, which generalize the notion of affine vectors in the same way that (conformal) Killing tensors generalize (conformal) Killing vectors. An identity for these tensors is proved, which gives the second derivative of the tensor in terms of the curvature tensor, generalizing a well-known identity for affine vectors. Additionally, the definition leads to a good definition of homothetic tensors. The inclusion relations between these types of tensors are exhibited. The relationship between affine symmetry tensors and solutions to the equation of geodesic deviation is clarified, again extending known results about Killing tensors.