Identification of Riemannian foliations on the tangent bundle via SODE structure

Document Type: Research Paper

Authors

Amirkabir University of Technology

Abstract

The geometry of a system of second order differential equations is the geometry of a semispray, which is a globally defined
vector field on TM. The metrizability of a given semispray is of special importance. In this paper, the metric associated with
the semispray S is applied in order to study some types of foliations on the tangent bundle which are compatible with SODE
structure. Indeed, sufficient conditions for the metric associated with the semispray S are obtained to extend to a bundle-like
metric for the lifted foliation on TM. Thus, the lifted foliation converts to a Riemanian foliation on the tangent space which
is adapted to the SODE structure. Particularly, the metrizability property of the semispray S is applied in order to induce
SODE structure on transversals. Finally, some equivalent conditions are presented for the transversals to be totally geodesic.

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Volume 38, Issue 3
September and October 2012
Pages 669-688
  • Receive Date: 26 December 2010
  • Revise Date: 10 April 2011
  • Accept Date: 10 April 2011