Surfaces of rotation with constant mean curvature in the direction of a unitary normal vector field in a Randers space (Q2880079)

This paper deals with hypersurfaces of constant mean curvature in a Randers space \((V^n, F_b)\) obtained by perturbing the standard Euclidean metric on \(V^n={\mathbb R}^n\) by a translation, where \(F=\alpha+\beta\) and \(b\) is the norm of the \(1\)-form \(\beta\). In Riemannian geometry, the study of hypersurfaces of constant mean curvature is an important topic with many results. However, the theory of submanifolds in a Finsler space is relatively more subtle. It is proved by \textit{Z. Shen} [Math. Ann. 311, No. 3, 549--576 (1998; Zbl 0921.53037)] that the mean curvature form always vanishes on the tangent bundle.NEWLINENEWLINEThe authors of this paper study hypersurfaces of constant mean curvature in the direction of a unitary normal vector field. They obtain the ordinary differential equation that characterizes the rotational surfaces \((V^3, F_b)\) of constant mean curvature in the direction of a unitary vector field, generating the results on the Euclidean spaces and some previous results of \textit{M. Souza} and \textit{K. Tenenblat} [Math. Ann. 325, No. 4, 625--642 (2003; Zbl 1066.53124)]. The authors also study the special case of round cylinders and show that they are surfaces of constant mean curvature in the direction of the unitary normal field.