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In [[mathematics]] and [[physics]], a '''recurrent tensor''', with respect to a [[connection (mathematics)|connection]] <math>\nabla</math> on a [[manifold]] ''M'', is a [[Tensor field|tensor]] ''T'' for which there is a [[differential form|one-form]] ''ω'' on ''M'' such that

In mathematics, a '''recurrent tensor''' with respect to a [[connection (mathematics)|connection]] <math>\nabla</math> on a [[manifold]] ''M'' is a [[Tensor field|tensor]] ''T'' for which there is a [[differential form|one-form]] ''ω'' on ''M'' such that

:<math>\nabla T = \omega\otimes T. \, </math>

==Examples==

===Parallel Tensors===

An Exampleexample for recurrent tensors[[Tensor field|tensor]]s are parallel tensors which are defined by

:<math>\nabla A = 0 </math>

with respect to some connection <math>\nabla</math>.

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and its property to be torsion-free.

Important tensors are recurrent vector fields such as parallelParallel vector fields (<math>\nabla X = 0</math>) which are importantexamples inof mathematicrecurrent research.tensors Athat resultfind forimportance recurrentin vectormathematical fieldsresearch. onFor example, a [[pseudo-Riemannian manifold]]if <math>(M,g) X </math> is thea following.recurrent Letnon-null <math>X</math>vector befield on a recurrent[[pseudo-Riemannian vectorfieldmanifold]] satisfying

:<math>\nabla X = \omega\otimes X </math>

for some closed [[one-form]] <math> \omega </math>. Now if <math>d\omega = 0 </math> (<math>\omega</math> closed), e.g. if the length of <math>X</math> is not vanishing,then X can be rescaled to a parallel vectorfieldsvector field.<ref>Alekseevsky, Baum (2008)</ref>. In particular, non -parallel, recurrent vector fields are lightlikenull vektorfieldsvector fields.

===Metric space===

Another example for a recurrent tensor appears in connection with [[Weyl structure|Weyl structures]]s. HistoricalHistorically, Weyl structures emergeemerged from considerationthe considerations of [[Hermann Weyl]] onwith regards to properties of parallel transport of vectors and their length .<ref>Weyl (1918)</ref>. By claimingdemanding that a manifold to have aan affine parallel transport in such a way that the manifold is locally looks like an [[affine space]], heit gotwas ashown special property forthat the induced connection to havehad a vanashingvanishing torsion tensor

:<math>T^\nabla(X,Y) = \nabla_XY-\nabla_YX - [X,Y] = 0</math>.

In additionAdditionally, he claimed that the manifold tomust have a specialparticular parallel transport of the metric or scale in every point, which does not leave length of single vectors untouched but fixesthe ratio of two parallel transported vectors is fixed. AThe corresponding connection <math>\nabla'</math>, which induces such a parallel transport than fulfillssatisfies

:<math>\nabla' g = \varphi \otimes g</math>

for some one-form <math>\varphi</math>. In particular is suchSuch a metric is a reccurentrecurrent tensor with respect to <math>\nabla'</math>. As a result, Weyl called athe resulting manifold <math>(M,g)</math> with affine connection <math>\nabla</math> and recurrent metric ''<math> g'' </math> a metric space. NowadaysIn thethis term metric space is used slightly more general. Accuratelysense, Weyl was not just referring to one metric but to the conformal structure defined by ''<math> g'' which can be motivated as follows:</math>.

Under the conformal changestransformation <math>g \rightarrow e^{\lambda}g</math>, the form <math>\phivarphi</math> changestransforms as <math>\varphi \rightarrow \varphi -d\lambda</math>. This induces a canonical map <math>F:[g] \rightarrow \Lambda^1(M)</math> on <math>(M, [g])</math> asdefined follows:by

:<math>F(e^\lambda g) := \varphi - d\lambda</math>,

where <math>[g]</math> is the conformal structure. <math>F</math> is called a Weyl structure ,<ref>Folland (1970)</ref>, which more generalygenerally is defined as a map with property

:<math>F(e^\lambda g) = F(g) - d\lambda</math>.

===Recurrent spacetime===

One more example of a recurrent tensor is the curvature tensor <math>\mathcal{R}</math> on a recurrent spacetime ,<ref>Walker (1948)</ref>, for which

:<math>\nabla \mathcal{R} = \omega \otimes \mathcal{R}</math>.

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==Literature==

*{{cite journal |author=Weyl, H. |title=Gravitation und Elektrizität |journal=Sitzungsberichte der preussPreuss. Akad. dD. Wiss. |year=1918 |pages=465-478}} Reprinted in ''Das Relativitätsprinzip: Eine Sammlung von Originalarbeiten zur Relativitätstheorie Einsteins'' (1923), Wiesbaden: Vieweg+Teubner Verlag, pp. 147–159, {{doi|10.1007/978-3-663-19510-8_11}}.

*{{cite journal

* [http://qjmath.oxfordjournals.org/cgi/reprint/os-20/1/135.pdf A.G. Walker: ''On parallel fields of partially null vector spaces''], The Quarterly Journal of Mathematics 1949, Oxford Univ. Press

| last = Walker | first = A. G.

* [http://qjmath.oxfordjournals.org/cgi/reprint/2/1/151.pdf E.M. Patterson: ''On symmetric recurrent tensors of the second order''], The Quarterly Journal of Mathematics 1950, Oxford Univ. Press

| doi = 10.1093/qmath/os-20.1.135

*[http://www.jstor.org/stable/1993404 J.-C. Wong: ''Recurrent Tensors on a Linearly Connected Differentiable Manifold''], Transactions of the American Mathematical Society 1961,

| journal = The Quarterly Journal of Mathematics

* [http://www.intlpress.com/JDG/archive/1970/4-1&2-145.pdf G.B. Folland: ''Weyl Manifolds''], J. Differential Geometry 1970

| mr = 33588

*{{cite book | author=D.V. Alekseevky, H. Baum| title = Recent developments in pseudo-Riemannian geometry | publisher=European Mathematical Society | year=2008 |isbn = 3-037-19051-5}}

| pages = 135–145

| series = Oxford Series

| title = On parallel fields of partially null vector spaces

| volume = 20

| year = 1949}}

*{{cite journal

| last = Patterson | first = E. M.

| doi = 10.1093/qmath/2.1.151

| journal = The Quarterly Journal of Mathematics

| mr = 42771

| pages = 151–158

| series = Second Series

| title = On symmetric recurrent tensors of the second order

| volume = 2

| year = 1951}}

*{{cite journal

| last = Wong | first = Yung-chow

| doi = 10.1090/S0002-9947-1961-0121751-2

| journal = Transactions of the American Mathematical Society

| jstor = 1993404

| mr = 121751

| pages = 325–341

| title = Recurrent tensors on a linearly connected differentiable manifold

| volume = 99

| year = 1961}}

*{{cite journal

| last = Folland | first = Gerald B.

| doi = 10.4310/jdg/1214429379

| journal = Journal of Differential Geometry

| mr = 264542

| pages = 145–153

| title = Weyl manifolds

| volume = 4

| year = 1970}}

*{{cite book | author=D.V. Alekseevky,|author2= H. Baum|author2-link= Helga Baum | title = Recent developments in pseudo-Riemannian geometry | publisher=European Mathematical Society | year=2008 |isbn = 978-3-03703719-19051051-57}}

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[[Category:Riemannian geometry]]

[[Category:Tensors]]

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