Recurrent tensor
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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
[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]Literature
[edit]- 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.