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Symmetry fields of geodesic vector field

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*Kalnitski Vyacheslav S.*

**E-mail:** ezryimc@mail.wplus.net
**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