Abstract
We study the structure of those vector fields on the tangent bundle of an arbitrary smooth manifold which commute with the geodesic vector field defined by an affine connection. The study is restricted to polylinear fields generated by a pair of symmetric pseudotensor fields of type (k, 1) and (k+1,1), k≥0, defined on the manifold. We establish an isomorphism between the space of infinitesimal automorphisms of fixed type and the space ℌk of the solutions of a partial differential equation generalizing the Jacobi equation for the infinitesimal automorphisms of the connection. It is shown that the spaces ℌk are finite-dimensional and form a graduated Lie algebra ℌ=⊕ ∞k=0 ℌk. These algebras are classified in the case of one-dimensional manifolds. It is proved that if the geodesic vector field is complete, then so are the automorphisms corresponding to covariant constant fields of type (1, 1). Bibliography: 5 titles.
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Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 222–244.
Translated by V. S. Kal’nitskii.
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Kal’nitskii, V.S. The algebra of generalized jacobifields. J Math Sci 91, 3476–3491 (1998). https://doi.org/10.1007/BF02434926
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DOI: https://doi.org/10.1007/BF02434926