Recurrent tensor

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

\nabla T = \omega\otimes T. \,


Parallel Tensors[edit]

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

\nabla A = 0

with respect to some connection \nabla.

If we take a pseudo-Riemannian manifold (M,g) then the metric g is a parallel and therefore recurrent tensor with respect to its Levi-Civita connection, which is defined via

\nabla^{LC} g = 0

and its property to be torsion-free.

Parallel vector fields (\nabla X = 0) are examples of recurrent tensors that find importance in mathematical research. For example, if  X is a recurrent non-null vector field on a pseudo-Riemannian manifold satisfying

\nabla X = \omega\otimes X

for some closed one-form  \omega , 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

T^\nabla(X,Y) = \nabla_XY-\nabla_YX - [X,Y] = 0.

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 \nabla' which induces such a parallel transport satisfies

\nabla' g = \varphi \otimes g

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

Under the conformal transformation g \rightarrow e^{\lambda}g, the form \varphi transforms as \varphi \rightarrow \varphi -d\lambda. This induces a canonical map F:[g] \rightarrow \Lambda^1(M) on (M, [g]) defined by

F(e^\lambda g) := \varphi - d\lambda,

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

F(e^\lambda g) = F(g) - d\lambda.

Recurrent spacetime[edit]

One more example of a recurrent tensor is the curvature tensor \mathcal{R} on a recurrent spacetime,[4] for which

\nabla \mathcal{R} = \omega \otimes \mathcal{R}.


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