Recurrent tensor

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

${\displaystyle \nabla T=\omega \otimes T.\,}$

Examples

Parallel Tensors

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

${\displaystyle \nabla A=0}$ 

with respect to some connection ${\displaystyle \nabla }$ .

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

${\displaystyle \nabla ^{LC}g=0}$ 

and its property to be torsion-free.

Important tensors are recurrent vector fields such as parallel vector fields (${\displaystyle \nabla X=0}$ ) which are important in mathematic research. A result for recurrent vector fields on a pseudo-Riemannian manifold ${\displaystyle (M,g)}$  is the following. Let ${\displaystyle X}$  be a recurrent vectorfield satisfying

${\displaystyle \nabla X=\omega \otimes X}$ 

for some one-form ${\displaystyle \omega }$ . Now if ${\displaystyle d\omega =0}$  (${\displaystyle \omega }$  closed), e.g. if the length of ${\displaystyle X}$  is not vanishing, X can be rescaled to a parallel vectorfields ^[1]. In particular non parallel, recurrent vector fields are lightlike vektorfields.

Metric space

Another example for a recurrent tensor appears in connection with Weyl structures. Historical Weyl structures emerge from consideration of Hermann Weyl on properties of parallel transport of vectors and their length ^[2]. By claiming a manifold to have a affine parallel transport in such a way that the manifold locally looks like an affine space he got a special property for the induced connection to have a vanashing torsion tensor

${\displaystyle T^{\nabla }(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]=0}$ .

In addition he claimed the manifold to have a special parallel transport of the metric or scale in every point, which does not leave length of single vectors untouched but fixes ratio of two parallel transported vectors. A connection ${\displaystyle \nabla '}$ , which induces such a parallel transport than fulfills

${\displaystyle \nabla 'g=\varphi \otimes g}$ 

for some one-form ${\displaystyle \varphi }$ . In particular is such a metric is a reccurent tensor with respect to ${\displaystyle \nabla '}$ . As a result Weyl called a manifold ${\displaystyle (M,g)}$  with affine connection ${\displaystyle \nabla }$  and recurrent metric g a metric space. Nowadays the term metric space is used slightly more general. Accurately Weyl was not just referring to one metric but to the conformal structure defined by g which can be motivated as follows:

Under conformal changes ${\displaystyle g\rightarrow e^{\lambda }g}$  the form ${\displaystyle \phi }$  changes as ${\displaystyle \varphi \rightarrow \varphi -d\lambda }$ . This induces a canonical map ${\displaystyle F:[g]\rightarrow \Lambda ^{1}(M)}$  on ${\displaystyle (M,[g])}$  as follows:

${\displaystyle F(e^{\lambda }g):=\varphi -d\lambda }$ ,

where ${\displaystyle [g]}$  is the conformal structure. ${\displaystyle F}$  is called a Weyl structure ^[3], which more generaly is defined as a map with property

${\displaystyle F(e^{\lambda }g)=F(g)-d\lambda }$ .

Recurrent spacetime

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

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

References

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

Literature

• Weyl, H. (1918). "Gravitation und Elektrizität". Sitzungsberichte der preuss. Akad. d. Wiss.: 465.
• A.G. Walker: On parallel fields of partially null vector spaces, The Quarterly Journal of Mathematics 1949, Oxford Univ. Press
• E.M. Patterson: On symmetric recurrent tensors of the second order, 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, J. Differential Geometry 1970
• D.V. Alekseevky, H. Baum (2008). Recent developments in pseudo-Riemannian geometry. European Mathematical Society. ISBN 3-037-19051-5.