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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 Letvector <math>X</math>field beon 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 vectorfieldvector field <ref>Alekseevsky, Baum (2008)</ref>. In particular, non -parallel, recurrent vector fields are lightlike 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 recurrent 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</math>. follows:
Under the conformal changestransformation <math>g \rightarrow e^{\lambda}g</math>, the form <math>\phi</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 generally is defined as a map with property
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