Recurrent tensor: Difference between revisions

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===Metric space===

Another example appears in connection with [[Weyl structure|Weyl structures]]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

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Under the conformal transformation <math>g \rightarrow e^{\lambda}g</math>, the form <math>\phi</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>.

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* [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

* [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

*[http://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. Differential Geometry 1970

*{{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}}

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[[Category:Riemannian geometry]]

[[Category:Tensors]]

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[[de:Rekurrenter Tensor]]