Recurrent tensor

In mathematics, a recurrent tensor, with respect to a connection formula_1 on a manifold "M", is a tensor "T" for which there is a one-form "ω" on "M" such that
Examples.
Parallel Tensors.
An example for recurrent tensors are parallel tensors which are defined by 
with respect to some connection formula_1.
If we take a pseudo-Riemannian manifold formula_5 then the metric "g" is a parallel and therefore recurrent tensor with respect to its Levi-Civita connection, which is defined via
and its property to be torsion-free.
Parallel vector fields (formula_7) are examples of recurrent tensors that find importance in mathematical research. For example, if formula_8 is a recurrent non-null vector field on a pseudo-Riemannian manifold satisfying
for some closed one-form formula_10, then X can be rescaled to a parallel vector field. In particular, non-parallel recurrent vector fields are null vector fields.
Metric space.
Another example appears in connection with Weyl structures. Historically, Weyl structures emerged from the considerations of Hermann Weyl with regards to properties of parallel transport of vectors and their length. By demanding that a manifold have an affine parallel transport in such a way that the manifold is locally an affine space, it was shown that the induced connection had a vanishing torsion tensor
Additionally, he claimed that the manifold must have a particular parallel transport in which the ratio of two transported vectors is fixed. The corresponding connection formula_12 which induces such a parallel transport satisfies
for some one-form formula_14. Such a metric is a recurrent tensor with respect to formula_12. As a result, Weyl called the resulting manifold formula_5 with affine connection formula_1 and recurrent metric formula_18 a metric space. In this sense, Weyl was not just referring to one metric but to the conformal structure defined by formula_18. 
Under the conformal transformation formula_20, the form formula_14 transforms as formula_22. This induces a canonical map formula_23 on formula_24 defined by
where formula_26 is the conformal structure. formula_27 is called a Weyl structure, which more generally is defined as a map with property
Recurrent spacetime.
One more example of a recurrent tensor is the curvature tensor formula_29 on a recurrent spacetime, for which