It was observed by R. Kusner and proved by J. Ratzkin that one can connect together two constant mean curvature surfaces having two ends with the same Delaunay parameter. This gluing procedure is known as a “end-to-end connected sum”. In this paper we generalize, in any dimension, this gluing procedure to construct new constant mean curvature hypersurfaces starting from some known hypersurfaces.

In this paper there are discussed the three-component distributions of affine space $A_{n+1}$. Functions $\left\{{ℳ}^{\sigma }\right\}$, which are introduced in the neighborhood of the second order, determine the normal of the first kind of $\mathscr{H}$-distribution in every center of $\mathscr{H}$-distribution. There are discussed too normals $\left\{{𝒵}^{\sigma }\right\}$ and quasi-tensor of the second order $\left\{{𝒮}^{\sigma }\right\}$. In the same way bunches of the projective normals of the first kind of the $ℳ$-distributions were determined in the differential neighborhood of the second and third order.

In this note all vectors and $\epsilon$-vectors of a system of $m\le n$ linearly independent contravariant vectors in the $n$-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation $F(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{m}{\to }u)={(detA)}^{\lambda }·A·F(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u)$ with $\lambda =0$ and $\lambda =1$, for an arbitrary pseudo-orthogonal matrix $A$ of index one and given vectors $\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u.$

There are four kinds of scalars in the $n$-dimensional pseudo-Euclidean geometry of index one. In this note, we determine all scalars as concomitants of a system of $m\le n$ linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation $F(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{m}{\to }u)=\varphi \left(A\right)·F(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u)$ using two homomorphisms $\varphi$ from a group $G$ into the group of real numbers ${ℝ}_0=\left(ℝ\setminus \left\}0\right\{,·\right)$.

In this note, there are determined all biscalars of a system of $s\le n$ linearly independent contravariant vectors in $n$-dimensional pseudo-Euclidean geometry of index one. The problem is resolved by finding a general solution of the functional equation $F(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{s}{\to }u)=\left(\text{sign}(detA)\right)F(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{s}{\to }u)$ for an arbitrary pseudo-orthogonal matrix $A$ of index one and the given vectors $\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{s}{\to }u$.

Se obtiene un nuevo método para obtener toros de Willmore en estructuras conformes de Kaluza-Klein sobre fibrados principales con fibra la circunferencia. Diversas aplicaciones de esta técnica son consideradas.

The original version of the article was published in Central European Journal of Mathematics, 2012, 10(5), 1733–1762, DOI: 10.2478/s11533-012-0091-x. Unfortunately, it contains a typographical error in display of (27). Here we display the correct version of the equations.