Bochner technique in real Finsler manifolds.
The non-linear connection associated to Cartan-Finsler connection is used to study the Bochner technique of real Finsler manifolds. Some results about the relations between the vectors and the Ricci curvature of real Finsler manifolds are obtained. De Rham developed a suit of conventional symbols and an efficient method in treating the Laplace operator calculus on manifolds. A sketch of this method for the case of Finsler manifolds and the Weitzenböck formula are given. Using non-linear connections of a Finsler manifold \(M\), the existence of local coordinates, normalized at a point \(x\) is proved, and the Laplace operator \(\Delta \) on 1-forms on \(M\) is defined by nonlinear connection and its curvature tensor. After proving the maximum principle theorem of Hopf-Bochner on \(M\), the Bochner type vanishing theorem of Killing vectors and harmonic 1-form are obtained.
scientific article; zbMATH DE number 1931034