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

JANYŠKA, Josef. Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold. Archivum Mathematicum. Brno: MU Brno, 2001, vol. 37, No 2, p. 143-160. ISSN 0044-8753.

Basic information

• Original name: Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold
• Authors: JANYŠKA, Josef.
• Edition: Archivum Mathematicum, Brno, MU Brno, 2001, 0044-8753.

Other information

• Original language: English
• Type of outcome: Article in a journal
• Field of Study: 10101 Pure mathematics
• Country of publisher: Czech Republic
• Confidentiality degree: is not subject to a state or trade secret
• WWW: URL
• RIV identification code: RIV/00216224:14310/01:00004221
• Organization unit: Faculty of Science
• Keywords in English: Poisson structure; pseudo-Riemannian manifold; natural operator
• Tags: natural operator, Poisson structure, pseudo-Riemannian manifold

Abstract

Let $M$ be a differentiable manifold with a pseudo-Riemannian metric $g$ and a linear symmetric connection $K$. We classify all natural 0-order vector fields and 2-vector fields on $TM$ generated by $g$ and $K$. We get that all natural vector fields are linear combinations of the vertical lift of $u\in T_xM$ and the horizontal lift of $u$ with respect to $K$. Similarlz all natural 2-vector fields are linear combinatins of two canonical 2-vector fields induced by $g$ and $K$. Conditions for natural vector fields and natural 2-vector fields to define a Jacobi or a Poisson structure on $TM$ are disscused.

Links

• GA201/99/0296, research and development project: Name: Diferenciální geometrie vyššího řádu

Investor: Czech Science Foundation, Differential geometry of higher order

• MSM 143100009, plan (intention): Name: Matematické struktury algebry a geometrie

Investor: Ministry of Education, Youth and Sports of the CR, Mathematical structures of algebra and geometry