Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 20, No. 1, pp. 79-81 (2004)

Geodesics on a central symmetric warped product manifold

N.M. Ben Youssif

University of Szeged, Hungary

Abstract: We consider a warped product Riemannian metric on the manifold $\mathbb{R}^n_0\times \mathbb{R}^1$ with the central symmetric warping function $\phi( x)=\| x\|^{-2}$ mapping \(\mathbb{R}^n_0\rightarrow \mathbb{R}^1.\) The orthogonal projections onto $\mathbb{R}^n_0$ of geodesics of this warped product manifold $\mathbb{R}^n_0\times_{\phi} \mathbb{R}^1$ are exactly the trajectories of the mechanical systems on $\mathbb{R}^n_0$ with potential function $c\phi( x)^{-1}$ with arbitrary constant $c$. In this case all bounded trajectories of the mechanical system are closed. We show that the projection of geodesics are conic sections and determine the parameters of these conic sections as functions of the initial values of geodesics.

Keywords: Warped product Riemannian manifold, geodesics, conic sections.

Classification (MSC2000): 53C20; 53C22

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