Atlas: Natural Poisson structures on the tangent bundle of a pseudo-Riemannian manifold by Josef Janyska

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International Congress on Differential Geometry in memory of Alfred Gray (1939-1998)
September 18-23, 2000
Universidad del País Vasco
Bilbao, Spain

Organizers
M. Fernández (chairman), R. Ibáñez, M. Macho-Stadler

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Natural Poisson structures on the tangent bundle of a pseudo-Riemannian manifold
by
Josef Janyska
Masaryk University of Brno, Czech Republic

Oral Communication

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

X(u)=a(h(u))u^V + b(h(u)) u^H,

where u^V is the vertical lift of u in T_xM, u^H is the horisontal lift of u with respect to K, h(u) = 1/2g(u, u) and a, b are smooth functions of one real variable.

All natural 2-vector fields are of the form

L(u) = g₁(h(u)) L(g, K) + g₂(h(u))u^V /\ u^H,

where L(g, K) is the canonical 2-vector field induced by g, K and g₁, g₂ are smooth functions of one real variable.

Finally we deduce conditions for (X, L) to define a Jacobi or a Poisson structures on TM. Conditions for L to be a Poisson 2-vector coincide with the results obtained in [Ja] for natural symplectic structures on TM.

[Ja] J. Janyska, Remarks on symplectic and contact 2-forms in relativistic theories, Bollettino U.M.I. (7) 9-B (1995), 587-616.

[KMS] I. Kolár, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag 1993.

Date received: May 30, 2000