Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold

@article{Janyška2001,
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 [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.},
author = {Janyška, Josef},
journal = {Archivum Mathematicum},
keywords = {Poisson structure; pseudo–Riemannian manifold; natural operator; Poisson structure; pseudo-Riemannian manifold; natural operator},
language = {eng},
number = {2},
pages = {143-160},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold},
url = {http://eudml.org/doc/248735},
volume = {037},
year = {2001},
}

TY - JOUR
AU - Janyška, Josef
TI - Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold
JO - Archivum Mathematicum
PY - 2001
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 037
IS - 2
SP - 143
EP - 160
AB - 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.
LA - eng
KW - Poisson structure; pseudo–Riemannian manifold; natural operator; Poisson structure; pseudo-Riemannian manifold; natural operator
UR - http://eudml.org/doc/248735
ER -