Recurrent tensor: Difference between revisions

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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

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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

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* [http://qjmath.oxfordjournals.org/cgi/reprint/2/1/151.pdf E.M. Patterson: ''On symmetric recurrent tensors of the second order'']{{dead link|date=May 2021|bot=medic}}{{cbignore|bot=medic}}, The Quarterly Journal of Mathematics 1950, Oxford Univ. Press

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

* [http://www.intlpress.com/JDG/archive/1970/4-1&2-145.pdf G.B. Folland: ''Weyl Manifolds''], J.Journal of Differential Geometry 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 = 3-03719-051-5}}