Hypersurfaces in Riemannian manifolds with torse-forming axes
Muhittin Evren Aydın, Adela Mihai, Cihan Özgür
TL;DR
The paper addresses hypersurfaces $N$ in a Riemannian manifold $M$ whose unit normal $U$ makes a constant angle with a globally defined torse-forming axis $\mathcal{V}$, i.e. $\langle U,\mathcal{V}\rangle$ is constant. It develops a tangential decomposition $\mathcal{V}=\sin\theta\,T+\cos\theta\,U$ for unit anti-torqued axes and derives key equations linking the axis, the unit tangent $T$, and the shape operator, then provides complete local classifications in the anti-torqued case for dimensions 3 and higher, including explicit warped- and twisted-product models and representative examples; it also treats the torqued-axis case, obtaining decompositions of $\mathcal{V}$ and its generative vector field $\mathcal{W}$ and showing how warped/twisted product structures arise under suitable conditions. The results extend constant angle submanifold theory to ambient spaces with a globally defined torse-forming axis and connect to classical models such as constant slope surfaces, cones in hyperbolic space, and cylinders in warped products. Collectively, the work provides concrete local descriptions and construction methods for constant angle hypersurfaces with varied torse-forming axes, enriching the geometric landscape of submanifolds in general Riemannian manifolds.
Abstract
In this paper, we study orientable hypersurfaces $N$ in Riemannian manifolds $(M,\langle , \rangle)$ for which the inner product $\langle U, \mathcal{V} \rangle$ is constant, where $U$ is the unit normal vector field to $N$ and $\mathcal{V}$ is a globally defined torse-forming vector field on $M$, called the axis of $N$. When $\mathcal{V}$ is a unit torse-forming vector field, $N$ becomes a constant angle hypersurface with axis $\mathcal{V}$, and we classify such hypersufaces. After that, the case when $\mathcal{V}$ is a torqued vector field is considered and a corresponding classification is given.