Earle–Hamilton fixed-point theorem: Difference between revisions
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In [[mathematics]] and [[mathematical physics]], a '''holonomic basis''' or '''coordinate basis''' for a [[differentiable manifold]] is a set of [[Basis (linear algebra)|basis]] [[vector field]]s {''e''<sub>''k''</sub>} such that ''some'' [[coordinate system]] {''x''<sup>''k''</sup>} exists for which | |||
:<math>e_k = {\partial \over \partial x^k}.</math> | |||
A local condition for a basis {''e''<sub>''k''</sub>} to be holonomic is that all mutual [[Lie derivative]]s vanish:<ref>{{citation |author1=Roger Penrose|author2=Wolfgang Rindler |title=Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields |publisher=Cambridge University Press |pages=197–199 }}</ref> | |||
:<math>[e_i,e_j]=0.</math> | |||
A basis that is not holonomic is called a non-holonomic or non-coordinate basis. | |||
==References== | |||
*{{reflist}} | |||
==See also== | |||
*[[Jet bundle]] | |||
*[[Tetrad formalism]] | |||
*[[Ricci calculus]] | |||
{{differential-geometry-stub}} | |||
[[Category:Differential geometry]] | |||
Revision as of 09:00, 20 August 2013
In mathematics and mathematical physics, a holonomic basis or coordinate basis for a differentiable manifold is a set of basis vector fields {ek} such that some coordinate system {xk} exists for which
A local condition for a basis {ek} to be holonomic is that all mutual Lie derivatives vanish:[1]
A basis that is not holonomic is called a non-holonomic or non-coordinate basis.
References
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See also
Template:Differential-geometry-stub
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