Combinatorial differential geometry and ideal Bianchi–Ricci identities II – the torsion case

Josef Janyška; Martin Markl

Archivum Mathematicum (2012)

  • Volume: 048, Issue: 1, page 61-80
  • ISSN: 0044-8753

Abstract

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This paper is a continuation of [2], dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without corrections.

How to cite

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Janyška, Josef, and Markl, Martin. "Combinatorial differential geometry and ideal Bianchi–Ricci identities II – the torsion case." Archivum Mathematicum 048.1 (2012): 61-80. <http://eudml.org/doc/246441>.

@article{Janyška2012,
abstract = {This paper is a continuation of [2], dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without corrections.},
author = {Janyška, Josef, Markl, Martin},
journal = {Archivum Mathematicum},
keywords = {natural operator; linear connection; torsion; reduction theorem; graph; natural operator; linear connection; reduction theorem},
language = {eng},
number = {1},
pages = {61-80},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Combinatorial differential geometry and ideal Bianchi–Ricci identities II – the torsion case},
url = {http://eudml.org/doc/246441},
volume = {048},
year = {2012},
}

TY - JOUR
AU - Janyška, Josef
AU - Markl, Martin
TI - Combinatorial differential geometry and ideal Bianchi–Ricci identities II – the torsion case
JO - Archivum Mathematicum
PY - 2012
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 048
IS - 1
SP - 61
EP - 80
AB - This paper is a continuation of [2], dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without corrections.
LA - eng
KW - natural operator; linear connection; torsion; reduction theorem; graph; natural operator; linear connection; reduction theorem
UR - http://eudml.org/doc/246441
ER -

References

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  1. Janyška, J., 10.1016/j.difgeo.2003.10.006, Differential Geom. Appl. 20 (2004), 177. (2004) Zbl1108.53016MR2038554DOI10.1016/j.difgeo.2003.10.006
  2. Janyška, J., Markl, M., 10.1515/advgeom.2011.017, Adv. Geom. 11 (3) (2011), 509–540. (2011) Zbl1220.53019MR2817591DOI10.1515/advgeom.2011.017
  3. Kolář, I., Michor, P. W., Slovák, J., Natural operations in differential geometry, Springer–Verlag, Berlin, 1993. (1993) Zbl0782.53013MR1202431
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  9. Markl, M., 10.1016/j.difgeo.2008.10.008, Differential Geom. Appl. 27 (2009), 257–278. (2009) Zbl1165.51005MR2503978DOI10.1016/j.difgeo.2008.10.008
  10. Markl, M., Shnider, S., Stasheff, J. D., Operads in Algebra, Topology and Physics, Mathematical Surveys and Monographs, vol. 96, Amer. Math. Soc., 2002. (2002) Zbl1017.18001MR1898414
  11. Markl, M., Voronov, A. A., PROPped up graph cohomology, Algebra, arithmetic, and geometry: In honor of Yu. I. Manin, vol. II, Birkhäuser Boston, Inc., Boston, MA, progr. math., 270 ed., 2009, pp. 249–281. (2009) Zbl1208.18008MR2641192
  12. Nijenhuis, A., Theory of the geometric object, Thesis, University of Amsterdam (1952). (1952) Zbl0049.22903MR0050364
  13. Nijenhuis, A., Natural bundles and their general properties. Geometric objects revisited, Differential geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 317–334. (1972) Zbl0246.53018MR0380862
  14. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, 1963. (1963) MR0152974
  15. Schouten, J. A., Ricci calculus, Berlin–Göttingen, 1954. (1954) Zbl0057.37803
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