Laplacians on the tangent bundle of Finsler manifold

Let M be a smooth manifold with a Finsler metric F and (g) over tilde be the naturally induced Riemann metric on the slit tangent bundle (M) over tilde. The Weitzenbock formulas of the horizontal Laplacian Delta(h) and the vertical Laplacian Delta(upsilon) are obtained in terms of the Cartan connection of (M, F). The relationship between the Hodge-Laplace operator Delta of (g) over tilde and the operators Delta(h), Delta(upsilon), Delta(mix) are investigated. As application, the relationship between the eigenvalues of Delta and Delta(h), Delta(upsilon) are established, and a Bochner-type vanishing theorem of horizontal differential form on (M) over tilde is obtained.