Geodesic
Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold
Archivum mathematicum, Tome 37 (2001) no. 2, pp. 143-160 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 [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 $R$. 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 $R$ 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.
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 [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 $R$. 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 $R$ 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.
Classification : 53C50, 53D17, 58A20, 58A32
Keywords: Poisson structure; pseudo–Riemannian manifold; natural operator 
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     author = {Jany\v{s}ka, Josef},
     title = {Natural vector fields and 2-vector fields on the tangent bundle of a {pseudo-Riemannian} manifold},
     journal = {Archivum mathematicum},
     pages = {143--160},
     year = {2001},
     volume = {37},
     number = {2},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2001_37_2_a5/}
}
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Janyška, Josef. Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold. Archivum mathematicum, Tome 37 (2001) no. 2, pp. 143-160. http://geodesic.mathdoc.fr/item/ARM_2001_37_2_a5/

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