Let \(M\) be an \(n\)-dimensional differentiable manifold of class \(C^{\infty }\), \(T(M\)) the tangent bundle over \(M\), \(p:V(M)\rightarrow M\) the fiber bundle of non-zero tangent vectors on \(M\) and \(p^{-1}:T(M)\rightarrow V(M)\) the fiber bundle induced by \(p\) on \(V(M)\). For any covariant differentiation \(\nabla \), a linear mapping \(\mu _{z}:T_{z}V(M)\rightarrow T(M)\) is defined by \(\mu _{z}(\widehat{X})\)=\(\nabla _{\widehat{X}}v\quad (z=(x,v)\in V(M))\). \(\nabla \) is called a regular connection if \(\mu _{z}\) is an isomorphism between \(V_{z}\) and \(T_{pz}\). Then, there exists a decomposition of the torsion tensor \(\tau \) in two tensors \(S\) and \(T\) and a decomposition of the curvature tensor \(\Omega \) into three curvature tensors \(R,P\) and \(Q\). Then, there exists a unique regular Euclidean connection of directions such that the torsion tensor \(S\) is vanishing and the torsion tensor \(T\) satisfies a symmetry condition. This characterization gives naturally the Euclidean connection defined by Cartan or by Berwald. If \(\pi \) is a connection of directions that defines the same splitting on the tangent fibre bundle of \(V(M)\) as the Cartan or Berwald connection of directions, then the geodesics and the flag curvature of the Cartan or Berwald connection of directions coincide with those of the connection \(\pi \). The Laplacian \(\Delta \) is defined as a sum of the horizontal Laplacian \({\overline{\Delta }}\) and the vertical Laplacian \(\dot{\Delta }\) associated with the horizontal covariant derivation in the Berwald connection. The author proves a very important formula linking the horizontal Laplacian \(\overline{\Delta }\) of a function from the fibre bundle of unitary tangent vectors to a Finslerian compact manifold without boundary to a symmetric 2-tensor and Finslerian curvature allowing him to estimate the eigenfunction \(\lambda \) of \({\overline{\Delta }}\). In particular case \(\lambda =nk\) where \(k\) is a positive constant and \(M\) is simply connected, then \(M\) is homeomorphic to an \(n\)-sphere. The manifolds with constant sectional curvature are also studied. It was proved that on a compact Finslerian manifold without boundary with non-zero constant sectional curvature in the Berwald connection every horizontally harmonic function on the fibre bundle \(W\) of unitary tangent vectors to the manifold is constant. If \(M\) is a Finslerian compact manifold without boundary and \(F^{0}(g_{t})\) is a deformation of the Finslerian metric which leaves the volume element of \(W\) invariant, then the author considers an integral \(I(g_{t})\) whose integrant \(\widetilde{H}\) is the trace of the second order vertical derivation of the directional Ricci curvature \(H\). The main result is: {For a Finslerian compact manifold dimension \(n\geq 2\), without boundary of the Finslerian metric which makes the integral \(I(g_{t})\) critical at the point \((t=0,g=g_{0}\in F^{0}(g_{t}))\) defines at this point a generalized Einstein manifold.} The author also studies the case when the scalar curvature \(\widetilde{H}\) is a non-positive constant.