Recurrent tensor
Recurrent tensors or recurrent tensor fields are used in the mathematical field of differential geometry.
Definition
In the differential geometry of a recurrent tensor is defined as follows: Let be a connection on a manifold M. A tensor A ( in terms of a tensor field ) is called recurrent with respect to the relationship, if there is a one-form on M such that
Examples
Parallel tensors
Example of recurrent tensors with respect to a relationship parallel tensors ().
An example of a parallel tensor is a (semi-) Riemannian metric with respect to its Levi -Civita connection. Let M be so, so as to define a manifold with metric g by the metric of the Levi- Civita connection, and from the definition follows then
Another example is recurrent vector fields, which can be here in special cases of recurrent vector fields parallel vector fields derived. Be time a semi Riemannian manifold and a rekurentes vector field with
It follows from (closed) that can rescale to a parallel vector field. In particular, you can turn any vector fields with non-zero length to rescale to a parallel vector field. Non- recurrent parallel vector fields are therefore particularly light-like.
Metric space
Another example of a recurrent tensor arises in connection with Weylstrukturen. Historically was the Weylstruktur from considerations of Hermann Weyl to properties of the parallel displacement of vectors and their length. The requirement to be able to describe a manifold locally affine, creates a condition on the associated with the affine parallel translation context. He must be torsion-free:
For additional parallel shift of the metric he called as a special property that, although not the length, but rather the aspect ratio remains obtained from parallel displaced vector fields. The thus defined context then satisfies the property
For a one-form. In particular, a recurrent tensor of the metric is to say with respect. The resulting manifold with affine connection and recurrent metric g called Weyl metric space now. In fact, Weyl considered not only a metric, but the conformal structure on g This can be motivated as follows:
Under a conformal change is transformed into the form whereby a picture on the manifold is induced conformal structure. These fixed it and and defined:
So satisfies the conditions of a Weylstruktur: