Matches in DBpedia 2015-10 for { ?s ?p "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 Anders Karlhede 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. Fortunately 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 }
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- Cartan–Karlhede_algorithm 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 Anders Karlhede 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. Fortunately 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.".