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Bi-Legendrian connections. (English) Zbl 1081.53028

In [Differential geometrical methods in mathematical physics, Proc. Conf. Aix-en-Provence and Salamanca 1979, Lect. Notes Math. 836, 153–166 (1980; Zbl 0464.58012)], H. Hess proved that a pair \(L,Q\) of complementary Lagrangian distributions of a symplectic manifold \(\left( M,\omega\right)\) uniquely determines a so-called bi-Lagrangian connection \(\nabla\) on \(M\). Moreover, if \(L,Q\) are involutive then \(\nabla\) is torsion free and flat along the leaves of the foliations \(L,Q\).
In the present paper, the author generalizes the notion of bi-Lagrangian connection to the notion of bi-Legendrian connection associated to a bi-Legendrian structure on an almost \(\mathcal{S}\)-manifold \(M^{2n+r}\) (a particular case of which, for \(r=1\), is a contact metric manifold). The author studies properties of this connection, in particular involving its torsion and curvature tensors. He also proves that if the bi-Legendrian connection is flat, then the bi-Legendrian structure is locally equivalent to the standard structure on \(\mathbb{R}^{2n+r}\) where \(2n+r\) is the dimension of the almost \(\mathcal{S}\)-manifold \(M\).

MSC:

53C12 Foliations (differential geometric aspects)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53B05 Linear and affine connections
57R30 Foliations in differential topology; geometric theory

Citations:

Zbl 0464.58012
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