On Infinitesimal Conformal Transformations of the Tangent Bundles with the Generalized Metric

Document Type : Research Article



Let  be an n-dimensional Riemannian manifold, and  be its tangent bundle with the lift metric. Then every infinitesimal fiber-preserving conformal transformation  induces an infinitesimal homothetic transformation on .  Furthermore,  the correspondence   gives a homomorphism of the Lie algebra of infinitesimal fiber-preserving conformal transformations on  onto the Lie algebra of infinitesimal homothetic transformations on , and the kernel of this homomorphism is naturally isomorphic onto the Lie algebra of infinitesimal isometries of .


[1]     M. T. K. Abbassi,  Note on the classification theorems of g-natural metrics on the tangent bundle of a Riemannian manifold (M, g),  Commnet. Math. Univ. Carolinae, 45 (4) (2004), 591-596.
[2]     H. Akbar-Zadeh, Transformations infinitesimals conformes des varietes finsleriennes  compactes,  Ann. Polon. Math., 36 (1979), 213-229.
[3]     M. Anastasiei, Locally conformal Kaehler structures on tangent manifold of a space form, Libertas Math., 19 (1999), 71-76.
[4]     I. Hasegawa and K. Yamauchi, Infinitesimal projective transformations on tangent bundles with lift connection, Scientiae Mathematicae Japonicae 52 (2003), 469-483.
[5]     R. Miron, and M. Anastasiei,, The Geometry of Lagrange spaces: Theory and applications, Kluwer Acad. Publ, FTPH, no.59,(1994).
[6]     R. Miron, and M. Anastasiei, Vector bundles and Lagrange spaces with application to Relativity. Geometry Balkan Press, Romania, (1981).
[7]     K. Yamauchi , On infinitesimal conformal transformations of the tangent bundles over Riemannian manifolds, Ann. Rep. Asahikawa. Med. Coll.Vol. 15. 1994.
[8]     K. Yano, The theory of Lie Derivatives and Its Applications, North Holland, (1957).
[9]     K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, New York, (1973).
[10]  K. Yano and S. Kobayashi,   Prolongations of tensor fields and connection to tangent bundle I, General theory, J. Math. Soc. Japan, 18 (1996), 194 -210.