Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Communications in Mathematics

Editor-in-Chief: Rossi, Olga

2 Issues per year


Mathematical Citation Quotient (MCQ) 2016: 0.28

Open Access
Online
ISSN
2336-1298
See all formats and pricing
More options …

Torsion and the second fundamental form for distributions

Geoff Prince
  • Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia
  • The Australian Mathematical Sciences Institute, c/o The University of Melbourne, Victoria 3010, Australia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-08-20 | DOI: https://doi.org/10.1515/cm-2016-0003

Abstract

The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one case generalises the traditional shape operator of Riemannian geometry.

Keywords: Torsion; second fundamental form; shape operator; integrable distributions

References

  • [1] A. Bejancu, H.R. Farran: Foliations and Geometric Structures. Springer (2006).Google Scholar

  • [2] M. Crampin, G.E. Prince: The geodesic spray, the vertical projection, and Raychaudhuri's equation. . Gen. Rel. Grav. 16 (1984) 675-689.Google Scholar

  • [3] M. Jerie, G.E. Prince: A generalised Raychaudhuri equation for secondorder differential equations. J. Geom. Phys. 34 (3) (2000) 226-241.CrossrefGoogle Scholar

  • [4] M. Jerie, G.E. Prince: Jacobi fields and linear connections for arbitrary second order ODE's.. J. Geom. Phys. 43 (4) (2002) 351-370.CrossrefGoogle Scholar

  • [5] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry. Wiley-Interscience, New York (1963).Google Scholar

  • [6] J. M. Lee: Riemannian manifolds: an introduction to curvature. Springer-Verlag, New York (1997).Google Scholar

About the article

Received: 2016-04-21

Accepted: 2016-05-19

Published Online: 2016-08-20

Published in Print: 2016-08-01


Citation Information: Communications in Mathematics, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2016-0003.

Export Citation

© 2016. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in