A Global Theory of Conformal Finsler Geometry

Abstract

The aim of the present paper is to establish a global theory of conformal changes in Finsler geometry. Under this change, we obtain the relationships between the most important geometric objects associated to $(M,L)$ and the corresponding objects associated to $(M,\tilde{L})$, $\widetilde{L}=e^{σ(x)} L$ being the Finsler conformal transformation. We have found explicit global expressions relating the two associated Cartan connections $\nabla$ and $\widetilde{\nabla}$, the two associated Berwald connections $D$ and $\widetilde{D}$ and the two associated Barthel connections $Γ$ and $\widetildeΓ$. The relationships between the corresponding curvature tensors have been also found. The relations thus obtained lead in turn to several interesting results. Among the results obtained, is a characterization of conformal changes, a characterization of homotheties, some conformal invariants and conformal $σ$-invariants. In addition, several useful identities have been found. Our global theory of conformal Finsler geometry is established within the Pull-back approach, making simultaneous use of the Klein-Grifone approach. This has been done via certain links we have found between both approaches (This shows that these two approaches to global Finsler geometry are not alternatives but rather complementary). Although our treatment is entirely global, the local expressions of the obtained results, when calculated, coincide with the existing classical local results.