Chapter III Isometries and affine vector fields on the unitary tangent fibre bundle

Abstract

Abstract This chapter is devoted to the study of infinitesimal isometries of a compact Finslerian manifold without boundary, and affine infinitesimal transformation of a regular linear connection of directions ([1], [1a], [1b]). We recall the calculus rules of Lie derivatives of a tensor field in the large sense and of a form of a regular linear connection of vectors. Let L be the Lie algebra of infinitesimal transformations of M. To an X ∈ L is associated a certain endomorphism AX of Tpz whose expression contains the torsion tensor T. Let A z(L) be the Lie algebra of endomorphisms of Tpz corresponding to the elements of L. We establish then a relation between A[X Y], AX, AY and the curvature of the linear connection, generalizing from the Riemannian case due to B. Kostant ([26], [1]). For the study of compact Finslerian manifolds we establish the divergence formulas for the horizontal 1-forms and for the vertical 1-forms on W(M) ([1],[2]). Next we study the 1-parameter group of infinitesimal transformations that leave invariant the splitting of the tangent bundle defined by Finslerian connection. We give a local characterization of isometries. In case M is compact and without boundary we prove the largest connected group of transformations that leave invariant the splitting defined by the Finslerian connection coincides with the largest group of isometries. We establish a formula linking the square of the vertical part of the lift of an isometry X on V(M) and the integral involving an expression of the flag curvature (R(X,u)u,X). If this form is negative definite the isometry group is finite. Finally, to every infinitesimal isometry X of a Finslerian manifold is associated an antisymmetric endomorphism whose square of the module modulo a divergence puts in evidence a quadratic form φ depending on two Ricci tensor Rij and Pij [1b]. We determine the conditions on them so that the isometry group of the manifold is finite. We study the particular case of Pij = 0 In paragraph §10 we give a characterization of affine infinitesimal transformation (respectively partial) of regular linear connections of vectors. In paragraph § 11 we show that the Lie derivative L(X) commutes with the covariant derivatives of two types ∇ and ∇• (respectively of type ∇) when X defines an affine infinitesimal transformation (respectively partial), and conversely. Let L be the Lie algebra of affine transformations of a generalized linear connection, and L ˜ its lift on V(M). The Lie algebra A z(L), corresponding to L is the Lie algebra of a connected group Kz(L) of linear transformations of Tpz. The study of this group is the objective of paragraphs § 12, 13 and 14. In the case when the Lie algebra L ˜ is transitive on V(M) we have a relation of inclusion σz ⊂ Kz L ⊂ N O σz where σz is the group of restricted homogeneous holonomy at z ∈ V(M) and NO(σz) indicates the passage to the connected normalizer [1]. The group Kz(L) has been introduced by B. Kostant [26] in the Riemannian case. In conclusion we also study the case of affine infinitesimal transformation of a Finslerian connection.