Tensor Generalizations of Affine Symmetry Vectors

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.