Combinatorial differential geometry and ideal Bianchi-Ricci identities

0362692 - MÚ 2012 RIV DE eng J - Journal Article
Janyška, J. - Markl, Martin
Combinatorial differential geometry and ideal Bianchi-Ricci identities.
Advances in Geometry. Roč. 11, č. 3 (2011), s. 509-540. ISSN 1615-715X. E-ISSN 1615-7168
R&D Projects: GA ČR GA201/08/0397
Institutional research plan: CEZ:AV0Z10190503
Keywords : Natural operator * linear connection * reduction theorem
Subject RIV: BA - General Mathematics
Impact factor: 0.338, year: 2011
http://www.degruyter.com/view/j/advg.2011.11.issue-3/advgeom.2011.017/advgeom.2011.017.xml

We apply the graph complex approach of [8] 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.
Permanent Link: http://hdl.handle.net/11104/0198945