About: Cartan–Karlhede algorithm

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The Cartan–Karlhede algorithm is a procedure for completely classifying and comparing Riemannian manifolds. Given two Riemannian manifolds of the same dimension, it is not always obvious whether they are locally isometric. Élie Cartan, using his exterior calculus with his method of moving frames, showed that it is always possible to compare the manifolds. Carl Brans developed the method further, and the first practical implementation was presented by in 1980.

Attributes	Values
rdf:type	anatomical structure
rdfs:label	Cartan–Karlhede algorithm (en)
rdfs:comment	The Cartan–Karlhede algorithm is a procedure for completely classifying and comparing Riemannian manifolds. Given two Riemannian manifolds of the same dimension, it is not always obvious whether they are locally isometric. Élie Cartan, using his exterior calculus with his method of moving frames, showed that it is always possible to compare the manifolds. Carl Brans developed the method further, and the first practical implementation was presented by in 1980. (en)
sameAs	Cartan–Karlhede algorithm
dbp:wikiPageUsesTemplate	dbt:Ill
Subject	Riemannian geometry Mathematical methods in general relativity
Link from a Wikipage to an external page	https://web.archive.org/web/20051023160408/http:/130.15.26.66/servlet/GRDB2.GRDBServlet
gold:hypernym	Procedure
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Cartan–Karlhede_algorithm?oldid=1061337626&ns=0
Wikipage page ID	2680495 (xsd:integer)
page length (characters) of wiki page	5826 (xsd:nonNegativeInteger)
Wikipage revision ID	1061337626 (xsd:integer)
Link from a Wikipage to another Wikipage	Fluid solution Frame fields in general relativity Null dust solution Carl H. Brans Riemannian geometry General relativity Metric tensor Petrov classification Curvature invariant Definite form Compact group Computer algebra system Covariant derivative Moving frame SHEEP (symbolic computation system) Exterior derivative Vacuum solution (general relativity) Vanishing scalar invariant spacetime Isometry Lorentz group Élie Cartan Lie group Riemann curvature tensor Riemannian manifold Mathematical methods in general relativity
has abstract	The Cartan–Karlhede algorithm is a procedure for completely classifying and comparing Riemannian manifolds. Given two Riemannian manifolds of the same dimension, it is not always obvious whether they are locally isometric. Élie Cartan, using his exterior calculus with his method of moving frames, showed that it is always possible to compare the manifolds. Carl Brans developed the method further, and the first practical implementation was presented by in 1980. The main strategy of the algorithm is to take covariant derivatives of the Riemann tensor. Cartan showed that in n dimensions at most n(n+1)/2 differentiations suffice. If the Riemann tensor and its derivatives of the one manifold are algebraically compatible with the other, then the two manifolds are isometric. The Cartan–Karlhede algorithm therefore acts as a kind of generalization of the Petrov classification. The potentially large number of derivatives can be computationally prohibitive. The algorithm was implemented in an early symbolic computation engine, SHEEP, but the size of the computations proved too challenging for early computer systems to handle. For most problems considered, far fewer derivatives than the maximum are actually required, and the algorithm is more manageable on modern computers. On the other hand, no publicly available version exists in more modern software. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Cartan–Karlhede_algorithm
is Wikipage redirect of	Cartan-Karlhede algorithm
is Link from a Wikipage to another Wikipage of	List of things named after Élie Cartan Cartan's equivalence method Cartan-Karlhede algorithm Curvature invariant List of contributors to general relativity GRTensorII Index of physics articles (C) Mathematics of general relativity
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Cartan–Karlhede_algorithm

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