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We consider hypersurfaces in the real Euclidean space $\mathbb{R}^{n+1}$ ($n\geq2$) which are relatively normalized. We give necessary and sufficient conditions a) for a surface of negative Gaussian curvature in $\mathbb{R}^3$ to be ruled,…

A Lorentz surface in the four-dimensional pseudo-Euclidean space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. In the present paper we obtain the complete classification of…

This paper deals with relative normalizations of skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$. In section 2 we investigate some new formulae concerning the Pick invariant, the relative curvature, the relative mean curvature…

Given a constant vector field $Z$ in Minkowski space, a timelike surface is said to have a canonical null direction with respect to $Z$ if the projection of $Z$ on the tangent space of the surface gives a lightlike vector field. In this…

In this article, we study spacelike and timelike rotational surfaces in a 3--dimensional de Sitter space $\mathbb{S}^3_1$ which are the orbit of a regular curve under the action of the orthogonal transformation of 4--dimensional Minkowski…

Meromorphic differentials on Riemann surfaces are said to be real-normalized if all their periods are real. Moduli spaces of real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles and residues…

In this paper, we study biconservative hypersurfaces in $\mathbb S^{n}$ and $\mathbb H^{n}$. Further, we obtain complete explicit classification of biconservative hypersurfaces in $4$-dimensional Riemannian space form with exactly three…

If $\alpha\in\r$, an $\alpha$-stationary surface in Euclidean space is a surface $\Sigma$ whose mean curvature $H$ satisfies $H(p)=\alpha |p|^{-2} \langle\nu,p\rangle$, $p\in\Sigma$. These surfaces generalize in dimension two a classical…

In the present paper we study normal transport surfaces in four-dimensional Euclidean space $\mathbb{E}^{4}$ which are the generalization of surface offsets in $\mathbb{E}^{3}$. We find some results of normal transport surfaces in…

Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of surfaces we study meridian surfaces with…

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…

Wintgen proved in [P. Wintgen, Sur l'in\'egalit\'e de Chen-Willmore, C. R. Acad. Sci. Paris, 288 (1979), 993--995] that the Gauss curvature $K$ and the normal curvature $K^D$ of a surface in the Euclidean 4-space $E^4$ satisfy $$K+|K^D|\leq…

Let $S$ be a minimal surface of general type with $p_{g}(S)=0$ and $K^{2}_{S}=4$. Assume the bicanonical map $\varphi$ of $S$ is a morphism of degree $4$ such that the image of $\varphi$ is smooth. Then we prove that the surface $S$ is a…

We consider the extrinsic geometry of surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the metric is degenerate, a surface normal cannot be unequivocally defined based on…

We consider Lie minimal surfaces, the critical points of the simplest Lie sphere invariant energy, in Riemannian space forms. These surfaces can be characterized via their Euler-Lagrange equations, which take the form of differential…

A meromorphic differential on a Riemann surface is said to be {\it real-normalized} if all its periods are real. Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds…

We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.

Wintgen ideal surfaces in E^4 form an important family of surfaces, namely surfaces with circular ellipse of curvature. Obviously, Wintgen ideal surfaces satisfy the pointwise equality K+K_N=H^2. In the present study we consider the Wintgen…

We consider a skew ruled surface $\Phi$ in the Euclidean space $E^{3}$ and relative normalizations of it, so that the relative normals at each point lie in the corresponding asymptotic plane of $\Phi$. We call such relative normalizations…

The study of quadric surfaces of revolution is a cornerstone of classical Euclidean geometry, but its extension to the three-dimensional sphere $\mathbb{S}^3$ has not been sufficiently explored. This article addresses this important gap by…