| Title:
|
Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold (English) |
| Author:
|
Janyška, Josef |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
37 |
| Issue:
|
2 |
| Year:
|
2001 |
| Pages:
|
143-160 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Poisson structure |
| Keyword:
|
pseudo–Riemannian manifold |
| Keyword:
|
natural operator |
| MSC:
|
53C50 |
| MSC:
|
53D17 |
| MSC:
|
58A20 |
| MSC:
|
58A32 |
| idZBL:
|
Zbl 1090.58007 |
| idMR:
|
MR1838411 |
| . |
| Date available:
|
2008-06-06T22:28:41Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107794 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| . |
