Recurrent tensor: Difference between revisions

Revision as of 14:31, 1 December 2010

Revision as of 20:51, 16 November 2011

Line 1:

{{context|orphandate = January 2010}}

{{multiple issues

|copyedit = April 2010

|orphan = January 2010

|context = January 2010

}}

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>

Line 11 ⟶ 7:

==Examples==

===Parallel Tensors===

An example for recurrent tensors[[Tensor field|tensor]]s are parallel tensors which are defined by

:<math>\nabla A = 0 </math>

with respect to some connection <math>\nabla</math>.

Line 30 ⟶ 26:

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>\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