In mathematics and physics, 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

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Parallel Tensors

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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

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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

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

 .

References

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

Literature

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  • Weyl, H. (1918). "Gravitation und Elektrizität". Sitzungsberichte der Preuss. Akad. D. Wiss.: 465.
  • A.G. Walker: On parallel fields of partially null vector spaces[dead link], The Quarterly Journal of Mathematics 1949, Oxford Univ. Press
  • E.M. Patterson: On symmetric recurrent tensors of the second order[dead link], The Quarterly Journal of Mathematics 1950, Oxford Univ. Press
  • J.-C. Wong: Recurrent Tensors on a Linearly Connected Differentiable Manifold, Transactions of the American Mathematical Society 1961,
  • G.B. Folland: Weyl Manifolds, Journal of Differential Geometry 1970
  • D.V. Alekseevky; H. Baum (2008). Recent developments in pseudo-Riemannian geometry. European Mathematical Society. ISBN 978-3-03719-051-7.