Torsion and the second fundamental form for distributions

Geoff Prince

Communications in Mathematics (2016)

  • Volume: 24, Issue: 1, page 23-28
  • ISSN: 1804-1388

Abstract

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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.

How to cite

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Prince, Geoff. "Torsion and the second fundamental form for distributions." Communications in Mathematics 24.1 (2016): 23-28. <http://eudml.org/doc/286718>.

@article{Prince2016,
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.},
author = {Prince, Geoff},
journal = {Communications in Mathematics},
keywords = {Torsion; second fundamental form; shape operator; integrable distributions},
language = {eng},
number = {1},
pages = {23-28},
publisher = {University of Ostrava},
title = {Torsion and the second fundamental form for distributions},
url = {http://eudml.org/doc/286718},
volume = {24},
year = {2016},
}

TY - JOUR
AU - Prince, Geoff
TI - Torsion and the second fundamental form for distributions
JO - Communications in Mathematics
PY - 2016
PB - University of Ostrava
VL - 24
IS - 1
SP - 23
EP - 28
AB - 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.
LA - eng
KW - Torsion; second fundamental form; shape operator; integrable distributions
UR - http://eudml.org/doc/286718
ER -

References

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  1. Bejancu, A., Farran, H.R., Foliations and Geometric Structures, 2006, Springer, (2006) Zbl1092.53021MR2190039
  2. Crampin, M., Prince, G.E., 10.1007/BF00767860, Gen. Rel. Grav., 16, 1984, 675-689, (1984) Zbl0541.53012MR0750379DOI10.1007/BF00767860
  3. Jerie, M., Prince, G.E., 10.1016/S0393-0440(99)00065-0, J. Geom. Phys., 34, 3, 2000, 226-241, (2000) MR1762775DOI10.1016/S0393-0440(99)00065-0
  4. Jerie, M., Prince, G.E., 10.1016/S0393-0440(02)00030-X, J. Geom. Phys., 43, 4, 2002, 351-370, (2002) MR1929913DOI10.1016/S0393-0440(02)00030-X
  5. Kobayashi, S., Nomizu, K., Foundations of Differential Geometry, 1, 1963, Wiley-Interscience, New York, (1963) Zbl0119.37502MR0152974
  6. Lee, J. M., Riemannian manifolds: an introduction to curvature, 1997, Springer-Verlag, New York, (1997) Zbl0905.53001MR1468735

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