*(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), $(n=dimM)\xb7$

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,Y$\in \mathcal{X}\left(TM\right)$; 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.