Groups of Transformations in Finslerian Saces, Basic Concepts of Finslerian Geometry: an Axiomatic Approach to Tensors

Infinitesimal transformations defining motions, affine motions, projective motions, conformal transformations and curvature collineations in various types of Finslerian spaces are discussed here. The notation and symbolism used is mainly based on [60] and author's works [24] ? [42]. The present article offers an exposition of the axiomatic definition of tensors and their further developments from this very standpoint. Various types of tensor sand their examples have been included. A systematic study of manifolds endowed with a metric defined by the positive fourth-root of a 4th degree differential form was considered by P. Finsler in 1918, after whom such manifolds were eventually named. Thereafter, several geometers: E. Cartan, L. Berwald, J. A. Schouten, J. Douglas, W. Barthel, H. Rund, A. Lchnerowicz, A. Kawaguchi, H. Busemann, A. Moór, K. Takano, S. S. Chern, M. S. Knebelman etc. explored this domain extensively. The first treatise on the subject (in English) was published by Rund in 1959. Main aspects of the theory are presented here more elegantly and briefly.