Almost contact metric structures defined by a symplectic structure over a~distribution

S. V. Galaev ; Yu. V. Shevtsova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 2, pp. 136-141.

Résumé

The distribution $D$ of an almost contact metric structure $(\varphi,\vec\xi,\eta,g)$ is an odd analogue of the tangent bundle. In the paper an intrinsic symplectic structure naturally associated with the initial almost contact metric structure is constructed. The interior connection defines the parallel transport of admissible vectors (i.e. vectors belonging to the distribution $D$) along admissible curves. Each corresponding extended connection is a connection in the vector bundle $(D,\pi,X)$ defined by the interior connection and by an endomorphism $N:D\to D$. The choice of the endomorphism $N:D\to D$ determines the properties of the extended connection, whence those of the almost contact metric structure appearing on the space $D$ of the vector bundle $(D,\pi,X)$. It is shown that similarly to the bundle $TTX$, the tangent bundle $TD$ due to the fixation of the connection over the distribution (and later also the $N$-extended connection, i.e. connection in the vector bundle $(D,\pi,X)$) is decomposable in the direct sum of the vertical and horizontal distributions. Thus on the distribution $D$ the (extended) almost contact metric structure is defined in a natural way. The properties of the extended structure are investigated. In particular, it is proved that the extended almost contact structure is almost normal if and only if the distribution $D$ is a distribution of zero curvature.

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