Symmetry fields of geodesic vector field

Kalnitski Vyacheslav S.


Abstract. The Lie algebra $y$ of the symmetry fields of the geodesic vector field on the tangent bundle of a smooth manifold allows the natural grading $y=\oplus_{i=0}^{\infty}h_i$, where the 0-rank fields $h_0$ correspond to the connection symmetry fields. One of the difficult questions is the completeness of the fields in the sence of that of solutions. Partly the problem is solved by the Kobayashi theorem, from which follows that $h_0$ is the Lie algebra of complete fields provided such is the geodesic field. Under the same condition the author found a subspace in $h_1$ consisting of complete fields. These can be constructed as $$ Y=A^i_sv^s\frac{\partial}{\partial x_i}+ (\nabla_kA^i_sv^sv^k-\omega^i_{ts}A^t_kv^sv^k)\frac{\partial}{\partial v_i}\,, $$ where ${\bf (x,v)}$ are standard coordinates of the tangent bundle of the manifold, $\omega$--- a connection form, $A$--- a covariant constant tensor field of type (1,1). For the flat connection on $R^n$ the infinite space of complete fields is presented.

AMSclassification. 58A05

Keywords. Tangent bundle, complete vector field, geodesic vector field