Summary
Recently we have studied the infinitesimal deformation in Finsler space [8] (1). Here we have obtained the necessary and sufficient condition for the infinitesimal change to become a motion in Finsler space. The condition is termed as the generalised Killing equation and the vector\(v^i \left( {x, \dot x} \right)\) occurring in this motion has been called the Killing vector. Various alternative forms of this equation have also been obtained. These forms are the generalisations of the corresponding equations obtained by Hokari [5], Knebelman [6] and Soós [10].
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References
Berwald L.,Unturschung der Krümmung allgemeiner metrischer Räume auf Grund des in inhen herrschenden Parallelismus, Math. Z. 25, 40–73 (1926).
Berwald L.,Über Finslersche und Cartansche Geometrie. IV.Projektivkrümmung allgemeiner affiner Räume und Finslersche Räume skalarer Krümmung, Ann. Math. (2) 48, 755–781 (1947).
Cartan E.,Les espaces de Finsler, Actualités Scientifiques 79, Paris (1934).
Eisenhart L. P.,Riemannian geometry, Princeton (1926).
Hokari S.,Winkeltreue Transformationen und Bewegungen im Finslerschen Raum, J. Fac. Sci. Hokkaido Univ., Ser. I. Math.5, 1–8 (1936).
Knebelman M. S.,Collineations and motions in generalised space, Amer. J. Math.51, 527–564 (1929).
Rund H.,The Differential Geometry of Finsler spaces, Springer-Verlag, Berlin, Göttingen, Heidelberg (1959).
Misra R. B. and Mishra R. S.,Lie derivatives of various geometric entities in Finsler Space, Forthcoming in Rev. de la Fac. Sci. Univ. Istanbul.
Sinha B. B.,Studies in Finsler spaces, Ph. D. Thesis, University of Gorakhpur (1962).
Soós G.,Über Gruppen von Affinitäten und Bewegungen in Finslerschen Räumen, Acta Math. Acad. Sci. Hungar.5, 73–84 (1954).
Thomas T. Y.,Differential invariants of generalised spaces, Cambridge (1934).
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Misra, R.B., Mishra, R.S. The Killing vector and the generalised Killing equation in Finsler space. Rend. Circ. Mat. Palermo 15, 216–222 (1966). https://doi.org/10.1007/BF02849437
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DOI: https://doi.org/10.1007/BF02849437