Recurrent tensor: Difference between revisions

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{{For|more|Riemannian geometry}}

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

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

for some closed [[one-form]] <math> \omega </math>, then X can be rescaled to a parallel vector field .<ref>Alekseevsky, Baum (2008)</ref>. In particular, non-parallel recurrent vector fields are null vector fields.

===Metric space===

Another example appears in connection with [[Weyl structure]]s. Historically, Weyl structures emerged from the considerations of [[Hermann Weyl]] with regards to properties of parallel transport of vectors and their length .<ref>Weyl (1918)</ref>. 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

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

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 <math>\nabla'</math> which induces such a parallel transport satisfies

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

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

Under the conformal transformation <math>g \rightarrow e^{\lambda}g</math>, the form <math>\phivarphi</math> transforms 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> defined 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 generally 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}}

{{DEFAULTSORT:Recurrent Tensor}}

[[Category:Riemannian geometry]]

[[Category:Tensors]]

[[de:Rekurrenter Tensor]]