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{{For|more|Riemannian geometry}}
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{{Technical|date=October 2021}}
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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]] ''
▲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>
==
===Parallel Tensors===
An example for recurrent [[Tensor field|tensor]]s are parallel tensors which are defined by
:<math>\nabla A = 0 </math>
with respect to some connection <math>\nabla</math>.
If we take a [[pseudo-Riemannian manifold]] <math>(M,g)</math> then the metric ''g'' is a parallel and therefore recurrent tensor with respect to its [[Levi-Civita connection]], which is defined via
:<math>\nabla^{LC} g = 0 </math>
and its property to be torsion-free.
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>\varphi</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>.
==References==
<references/>
==Literature==
*{{cite journal |author=Weyl, H. |title=Gravitation und Elektrizität |journal=Sitzungsberichte der Preuss. Akad. D. 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
| last = Walker | first = A. G.
| doi = 10.1093/qmath/os-20.1.135
*{{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}}▼
| journal = The Quarterly Journal of Mathematics
| mr = 33588
| 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
{{DEFAULTSORT:Recurrent Tensor}}
[[Category:Riemannian geometry]]
[[Category:Tensors]]
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