Previous |  Up |  Next


MSC: 53B05, 53C05, 58A10
Torsion; second fundamental form; shape operator; integrable distributions
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.
[1] Bejancu, A., Farran, H.R.: Foliations and Geometric Structures. 2006, Springer, MR 2190039 | Zbl 1092.53021
[2] Crampin, M., Prince, G.E.: The geodesic spray, the vertical projection, and Raychaudhuri's equation. Gen. Rel. Grav., 16, 1984, 675-689, DOI 10.1007/BF00767860 | MR 0750379 | Zbl 0541.53012
[3] Jerie, M., Prince, G.E.: A generalised Raychaudhuri equation for second–order differential equations. J. Geom. Phys., 34, 3, 2000, 226-241, DOI 10.1016/S0393-0440(99)00065-0 | MR 1762775
[4] Jerie, M., Prince, G.E.: Jacobi fields and linear connections for arbitrary second order ODE's. J. Geom. Phys., 43, 4, 2002, 351-370, DOI 10.1016/S0393-0440(02)00030-X | MR 1929913
[5] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. 1, 1963, Wiley-Interscience, New York, MR 0152974 | Zbl 0119.37502
[6] Lee, J. M.: Riemannian manifolds: an introduction to curvature. 1997, Springer-Verlag, New York, MR 1468735 | Zbl 0905.53001
Partner of
EuDML logo