A Framed f(3,-1) Structure on the Cotangent Bundle of a Hamilton Space

Girtu, Manuela, (2004) A Framed f(3,-1) Structure on the Cotangent Bundle of a Hamilton Space. Bulletin of the Malaysian Mathematical Sciences Society, 27 (2). pp. 161-168. ISSN 0126-6705

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Affiliations

University of Bacau, Faculty of Sciences

Abstract

For the cotangent bundle $(T \ast M, \tau^\ast , M)$ of a smooth manifold $M$ , the kernel of a differential $\tau_\ast ^\ast$ of the projection $\tau^\ast$ defines the vertical subbundle $VT \ast M$ of the bundle $(TT \ast M, \tau_{t \ast M}, T \ast M)$ . A supplement $HT \ast M$ of it is called a horizontal subbundle or a nonlinear connection on $M$ , [6,7]. The direct decomposition $TT \ast M = HT \ast M \oplus VT \ast M$ gives rise to a natural almost product structure $P$ on the manifold $T \ast M$ . A general method to associate to $P$ a framed $f(3,-1)$ -structure of any corank is pointed out. This is similar to that given by us in [2] for the tangent bundle of a Lagrange space. When we endow $M$ with a regular Hamiltonian $H$ and use as the nonlinear connection that canonically induced by $H$ , a framed $f(3,-1)$ -structure $P_2$ of corank $2$ naturally appears on $T \ast M$ . This reduces to that found by us in [3] when $H = K^2$ , for $K$ the fundamental function of a Cartan space $K^n =(M,K)$ . Then we show that on some conditions for $H$ the restriction of $P_2$ to the submanifold $H = 1$ of $T_0^\ast M$ provides an almost paracontact structure on this submanifold. The conditions taken on $H$ hold for the $\varphi$ -Hamiltonians introduced by us in [4] as well as for $H = K^2$ . The idea of this study has the origin in the paper [1] of M. Anastasiei.

Item Type:

Journal

Keywords:

cotangent bundle, framed f(3,-1)-structures, Hamilton spaces

Subjects:

Q Science, Computer Science

ID Code:

1305