Lagrange geometry on tangent manifolds. (English) Zbl 1045.53021

The Lagrange spaces, which are a generalization of the Finsler spaces, were introduced by J. Kern [Arch. Math. 25, 438–443 (1974; Zbl 0297.53035)] in order to geometrize a fundamental concept in mechanics, that of the Lagrangian. A regular Lagrangian is a differentiable function defined on the tangent bundle of a differentiable manifold \(M\) such that its Hessian with respect to the tangential coordinates is nondegenerate. Lagrange geometry, essentially, is the study of geometric objects and constructions that are transversal to the vertical foliation of the tangent bundle and are associated with a Lagrangian. In this paper, the author studies locally Lagrange manifolds, a generalization which consists of replacing the tangent bundle by a general tangent manifold, i. e., a \(2n\)-dimensional manifold \(M\) with a tensor field \(S\) of type \((1,1)\) such that \(S^2=0\), \(\operatorname{im} S=\ker S\) and integrable, that is, locally \(S\) looks like the vertical twisting homomorphism of a tangent bundle. Also, the Lagrangian is replacing by a family of compatible, local, Lagrangian functions. The locally Lagrange-symplectic manifolds, studied by the author in [I. Vaisman, Geom. Dedicata 74, No. 1, 79–89 (1999; Zbl 0942.53048)], are an important particular case of locally Lagrange manifols. The author gives several examples and obtains the cohomological obstructions to globalization. The last section is a extension of the connections used in Finsler and Lagrange geometry, which is given in an index-free presentation.


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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