Metrization of connections with regular curvature

Alena Vanžurová

Archivum Mathematicum (2009)

  • Volume: 045, Issue: 4, page 325-333
  • ISSN: 0044-8753

Abstract

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We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively existence of positive definite metrics compatible with a real analytic connection with regular curvature tensor on an analytic connected and simply connected manifold, and to construct the family of compatible metrics (determined up to a scalar multiple) in the affirmative case. We also breafly touch related problems concerning geodesic mappings and projective structures.

How to cite

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Vanžurová, Alena. "Metrization of connections with regular curvature." Archivum Mathematicum 045.4 (2009): 325-333. <http://eudml.org/doc/250560>.

@article{Vanžurová2009,
abstract = {We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively existence of positive definite metrics compatible with a real analytic connection with regular curvature tensor on an analytic connected and simply connected manifold, and to construct the family of compatible metrics (determined up to a scalar multiple) in the affirmative case. We also breafly touch related problems concerning geodesic mappings and projective structures.},
author = {Vanžurová, Alena},
journal = {Archivum Mathematicum},
keywords = {manifold; linear connection; metric; pseudo-Riemannian geometry; manifold; linear connection; metric; pseudo-Riemannian geometry},
language = {eng},
number = {4},
pages = {325-333},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Metrization of connections with regular curvature},
url = {http://eudml.org/doc/250560},
volume = {045},
year = {2009},
}

TY - JOUR
AU - Vanžurová, Alena
TI - Metrization of connections with regular curvature
JO - Archivum Mathematicum
PY - 2009
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 045
IS - 4
SP - 325
EP - 333
AB - We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively existence of positive definite metrics compatible with a real analytic connection with regular curvature tensor on an analytic connected and simply connected manifold, and to construct the family of compatible metrics (determined up to a scalar multiple) in the affirmative case. We also breafly touch related problems concerning geodesic mappings and projective structures.
LA - eng
KW - manifold; linear connection; metric; pseudo-Riemannian geometry; manifold; linear connection; metric; pseudo-Riemannian geometry
UR - http://eudml.org/doc/250560
ER -

References

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  1. Berger, M., A Panoramic View of Riemannian Geometry, Springer, Berlin, Heidelberg, New York, 2003. (2003) Zbl1038.53002MR2002701
  2. Borel, A., Lichnerowicz, A., Groupes d’holonomie des variétés riemanniennes, C. R. Acad. Sci. Paris 234 (1952), 1835–1837. (1952) Zbl0046.39801MR0048133
  3. Kobayashi, S., Nomizu, K., Foundations of Differential Geometry I, II, Wiley-Intersc. Publ., New York, Chichester, Brisbane, Toronto, Singapore, 1991. (1991) 
  4. Kowalski, O., 10.1007/BF01110924, Math. Z. 125 (1972), 129–138. (1972) Zbl0234.53024MR0295250DOI10.1007/BF01110924
  5. Kowalski, O., Metrizability of affine connections on analytic manifolds, Note di Matematica 8 (1) (1988), 1–11. (1988) Zbl0699.53038MR1050506
  6. Mikeš, J., 10.1007/BF02365193, J. Math. Sci. 78 (1996), 311–333. (1996) MR1384327DOI10.1007/BF02365193
  7. Mikeš, J., Kiosak, V., Vanžurová, A., Geodesic mappings of manifolds with affine connection, Palacký University, Olomouc (2008). (2008) Zbl1176.53004MR2488821
  8. Schmidt, B. G., 10.1007/BF01661152, Commun. Math. Phys. 29 (1973), 55–59. (1973) MR0322726DOI10.1007/BF01661152
  9. Vanžurová, A., Metrization problem for linear connections and holonomy algebras, Arch. Math. (Brno) 44 (2008), 339–349. (2008) Zbl1212.53021MR2501581
  10. Vilimová, Z., The problem of metrizability of linear connections, Master's thesis, 2004, (supervisor: O. Krupková). (2004) 

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