Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold
Janyška, Josef
Archivum Mathematicum, Tome 037 (2001), p. 143-160 / Harvested from 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.

Publié le : 2001-01-01
Classification:  53C50,  53D17,  58A20,  58A32 
@article{107794,
     author = {Josef Jany\v ska},
     title = {Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold},
     journal = {Archivum Mathematicum},
     volume = {037},
     year = {2001},
     pages = {143-160},
     zbl = {1090.58007},
     mrnumber = {1838411},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107794}
}

Janyška, Josef. Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold. Archivum Mathematicum, Tome 037 (2001) pp. 143-160. http://gdmltest.u-ga.fr/item/107794/

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