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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 $n=\left(M,L\right)$, 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), $\left(n=dimM\right)·$

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 $c\left(PX,PY\right)=-g$ c(X,Y), for all X,Y$\in 𝒳\left(TM\right)$; such that the pair (g c,P) is an almost Kählerian hyperbolic structure on TM. The Levi-Civita connection $\nabla$ 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:
 53C15 Differential geometric structures on manifolds 53B30 Lorentz metrics, indefinite metrics