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A pseudo-Riemannian structure in Lagrange geometry. (English) Zbl 0649.53018

Starting from some known results in the geometry of Lagrange spaces L n=(M,L), the author defines in the present paper the notion of complete lift g c of the metric g of L n to the total space TM of the tangent bundle to the manifold M. g c has a remarkable form and it is a pseudo- Riemannian metric on TM of signature (n,n), (n=dimM)·

He shows that g c together with the almost product structure P defined on TM by the horizontal distribution HTM of L n and the vertical distribution VTM have the property g c(PX,PY)=-g c(X,Y), for all X,Y𝒳(TM); such that the pair (g c,P) is an almost Kählerian hyperbolic structure on TM. The Levi-Civita connection of g c and its curvature tensor are expressed in the local adapted frame to the distribution HTM and VTM in detail. The Bianchi identities are also established.

Reviewer: R.Miron

MSC:
53C15Differential geometric structures on manifolds
53B30Lorentz metrics, indefinite metrics