Combinatorial differential geometry and ideal Bianchi–Ricci identities

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Authors

JANYŠKA Josef MARKL Martin

Year of publication 2011
Type Article in Periodical
Magazine / Source Advances in Geometry
MU Faculty or unit

Faculty of Science

Citation
Web http://www.degruyter.com/view/j/advg.2011.11.issue-3/advgeom.2011.017/advgeom.2011.017.xml?format=INT
Doi http://dx.doi.org/10.1515/advgeom.2011.017
Field General mathematics
Keywords Natural operator; linear connection; reduction theorem; graph
Description We apply the graph complex approach of~\cite{markl:na} to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe the size of the space of such operators and prove the existence of an `ideal' basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi--Ricci identities without the correction terms. The proofs given in this paper combine the classical methods of normal coordinates with the graph complex method.
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