Infinitesimal Transformation in a Finsler Space

Abstract

This paper has been divided into three sections, of which the first section is introductory, and the second section deals with normal projective infinitesimal transformation and curvature collineation in a Finsler space. In this section we give the following definitions: affine motion, normal projective curvature collineation, Ricci normal projective curvature collineation and infinitesimal normal projective transformation, also we have derived results in the form of Lie derivatives of normal projective curvature tensor 𝑁𝑗𝑘ℎ𝑖 and of the Ricci tensor 𝑁𝑘ℎ and in this continuation we have derived certain more results the projective deviation tensor and its Lie derivatives. After these observations we have derived result in the form of theorems telling as to what will happen to the covariant vector fields 𝑏𝑗(𝑥,𝑥 ) and 𝑑𝑟(𝑥,𝑥 ) when the infinitesimal normal projective transformation 𝑥 𝑖 = 𝑥𝑖 + 𝑣𝑖 (𝑥)𝑑𝑡 defines an affine as well as a non-affine motion and also derived the results which will hold when the infinitesimal normal projective point transformation 𝑥 𝑖 = 𝑥𝑖 + 𝑣𝑖 (𝑥)𝑑𝑡 defines a normal projective curvature and normal projective Ricci collineations. In this continuation, we have also derived results which will hold good if the infinitesimal normal projective transformation 𝑥 𝑖 = 𝑥𝑖 + 𝑣𝑖 (𝑥)𝑑𝑡 itself is affine and non-affine and also we have derived if the Finsler space under consideration is symmetric. The third and the last section we have study of infinitesimal projective transformation with special reference to Cartan’s connection Γ𝑗𝑘∗𝑖(𝑥,𝑥 ) here the previous section we have taken in the form of normal projective connection 𝛱𝑗𝑘𝑖(𝑥,𝑥 ) the two connection coefficients are quite different so results will be different. After these observations we have derived results in the form of theorems if the infinitesimal transformation 𝑥 𝑖 = 𝑥𝑖 + 𝑣𝑖 (𝑥)𝑑𝑡 defines an affine motion then in such a case the vector fields 𝑏𝑗(𝑥,𝑥 ) and 𝑑𝑙(𝑥,𝑥 ) should separately vanish. In this continuation we have also derived the relationships which will hold when the infinitesimal transformation 𝑥 𝑖 = 𝑥𝑖 + 𝑣𝑖 (𝑥)𝑑𝑡 defines Cartan’s curvature collineation as well as Cartan’s Ricci collineation. In the last we have derived the relationships which will hold when the infinitesimal transformation = 𝑥𝑖 + 𝑣𝑖 (𝑥)𝑑𝑡 is non-affine and affine one in a symmetric Finsler space.

Keywords: Normal projective infinitesimal transformation, Normal projective curvature collineation, Ricci normal projective curvature collineation, Lie-derivative, Cartan’s curvature collineation, Cartan’s Ricci collineation, Affine and non- affine motion, Symmetric Finsler space