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NATURAL VECTOR FIELDS AND 2-VECTOR FIELDS ON THE TANGENT BUNDLE OF A PSEUDO-RIEMANNIAN
MANIFOLD

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*Josef Janyska*

**Address.** Department of Mathematics, University of Masaryk Janackovo
nam 2a, 662 95 Brno, CZECH REPUBLIC
**E-mail:** janyska@math.muni.cz

**Abstract.** Let $M$ be a differentiable manifold with a pseudo-Riemannian
metric $g$ and a linear symmetric connection $K$. We classify all natural
(in the sense of \cite{KMS}) 0-order vector fields and 2-vector fields
on $TM$ generated by $g$ and $K$. We get that all natural vector fields
are of the form $$ E(u)=\alpha(h(u))\, u^H + \beta(h(u))\, u^V\,, $$ where
$u^V$ is the vertical lift of $u\in T_xM$, $u^H$ is the horizontal lift
of $u$ with respect to $K$, $h(u)= 1/2 g(u,u)$ and $\alpha,\beta$ are smooth
real functions defined on $\Bbb R$. All natural 2-vector fields are of
the form $$ \Lambda(u) = \gamma_1(h(u))\, \Lambda(g,K) + \gamma_2(h(u))\,u^H\wedge
u^V\,, $$ where $\gamma_1$, $\gamma_2$ are smooth real functions defined
on $\Bbb R$ and $\Lambda(g,K)$ is the canonical 2-vector field induced
by $g$ and $K$. Conditions for $(E,\Lambda)$ to define a Jacobi or a Poisson
structure on $TM$ are disscused.

**AMSclassification.** Primary 58A32; Secondary 53C50, 58A20, 53D17

**Keywords.** Poisson structure, pseudo-Riemannian manifold, natural
operator.