Recurrent tensor: Difference between revisions

Revision as of 16:30, 7 May 2010

Revision as of 18:01, 12 June 2010

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===Metric space===

Another example for a recurrent tensor appears in connection with [[Weyl structure|Weyl structures]]s. Historical Weyl structures emerge from consideration of [[Hermann Weyl]] on properties of parallel transport of vectors and their length <ref>Weyl (1918)</ref>. By claiming a manifold to have a affine parallel transport in such a way that the manifold locally looks like an [[affine space]] he got a special property for the induced connection to have a vanashing torsion tensor

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

In addition he claimed the manifold to have a special parallel transport of the metric or scale in every point, which does not leave length of single vectors untouched but fixes ratio of two parallel transported vectors. A connection <math>\nabla'</math>, which induces such a parallel transport than fulfills

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

for some one-form <math>\varphi</math>. In particular is such a metric is a reccurentrecurrent tensor with respect to <math>\nabla'</math>. As a result Weyl called a manifold <math>(M,g)</math> with affine connection <math>\nabla</math> and recurrent metric ''g'' a metric space. Nowadays the term metric space is used slightly more general. Accurately Weyl was not just referring to one metric but to the conformal structure defined by ''g'' which can be motivated as follows:

Under conformal changes <math>g \rightarrow e^{\lambda}g</math> the form <math>\phi</math> changes 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> as follows:

:<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 generalygenerally is defined as a map with property

:<math>F(e^\lambda g) = F(g) - d\lambda</math>.

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[[Category:Riemannian geometry]]

[[Category:Tensors]]

{{geometry-stub}}