arXiv subjects
Publications and source records attributed to Muhittin Evren Aydın.
We investigate the geometry of plane curves in $\mathbb{R}^2$ endowed with a canonical linear connection. By introducing the notions of the tangential and geodesic curvatures, we prove the existence and uniqueness theorems for curves with prescribed curvatures. We find all curves with constant curvature and we generalize the curves whose curvatures are linear functions of the arc length parameter.
In this paper, we introduce the notion of a prescribed angle hypersurface in a Riemannian manifold associated with a pair $(\mathcal{V},θ)$, where $\mathcal{V}$ is a unit vector field along the hypersurface and $θ$ denotes the angle between $\mathcal{V}$ and the unit normal vector field of the hypersurface. We study such hypersurfaces in case $\mathcal{V}$ is a torse-forming vector field. In the particular $3$-dimensional case, we determine the intrinsic and extrinsic curvatures of these hypersurfaces in terms of the prescribed angle and the potential function of $\mathcal{V}$. Using this, we classify prescribed angle surfaces under suitable assumptions.
In this paper, we first study invariants of curves parametrized by a real variable in the dual plane $\mathbb{D}^2$ under equiaffine transformations. We then obtain explicit equations for all curves in $\mathbb{D}^2$ whose equiaffine curvature is a dual constant. In particular, we prove that when the equiaffine curvature is a pure real constant, both the real and dual parts of the curve in $\mathbb{D}^2$ are quadratic curves. In addition, we provide a complete classification of spacelike and timelike curves parametrized by a real variable in the dual Lorentz--Minkowski plane $\mathbb{D}^2_1$ whose curvature is a dual constant.
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.