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

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.