Recurrent tensor

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In mathematics, a recurrent tensor, with respect to a connection on a manifold M, is a tensor T for which there is a one-form ω on M such that

Examples[edit]

Parallel Tensors[edit]

An example for recurrent tensors are parallel tensors which are defined by

with respect to some connection .

If we take a pseudo-Riemannian manifold 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 () are examples of recurrent tensors that find importance in mathematical research. For example, if is a recurrent non-null vector field on a pseudo-Riemannian manifold satisfying

for some closed one-form , then X can be rescaled to a parallel vector field.[1] In particular, non-parallel recurrent vector fields are null vector fields.

Metric space[edit]

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.[2] 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 which induces such a parallel transport satisfies

for some one-form . Such a metric is a recurrent tensor with respect to . As a result, Weyl called the resulting manifold with affine connection and recurrent metric a metric space. In this sense, Weyl was not just referring to one metric but to the conformal structure defined by .

Under the conformal transformation , the form transforms as . This induces a canonical map on defined by

,

where is the conformal structure. is called a Weyl structure,[3] which more generally is defined as a map with property

.

Recurrent spacetime[edit]

One more example of a recurrent tensor is the curvature tensor on a recurrent spacetime,[4] for which

.

References[edit]

  1. ^ Alekseevsky, Baum (2008)
  2. ^ Weyl (1918)
  3. ^ Folland (1970)
  4. ^ Walker (1948)

Literature[edit]