We start with a generic n-gon Q_0 with vertices q_{j,0} (j = 0, dots , n-1) in the d-dimensional Euclidean space {mathbb {E}}^d. Additionally, m+1 real numbers u_0, ldots , u_m in {mathbb {R}} , (m < n) with sum _{mu = 0}^m u_mu = 1 are given. From these initial data we iteratively define generations of n-gons Q_k in {mathbb {E}}^d for k in {mathbb {N}} with vertices q_{j,k}:= sum _{mu = 0}^m u_mu , q_{j+mu , k-1}. We can show that this affine iteration generally regularizes in an affine sense.

In this paper, using the representation formula of cone curves and solving the Riccati equations respected to the structure function of cone curve, we give the classifications of some cone curves in Minkowski 3-space.

In this paper, a characterization of the projective space PG(4, n) in terms of finite planar spaces is given.

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the characterization of frontal singularities found by Kokubu, Rossman, Saji, Umehara, Yamada, and Fujimori, Saji, Umehara, Yamada, and found by the first author related to the procedure of openings of singularities.

In this paper, we give a lower bound estimate for the diameter of a Lagrangian self-shrinker in a gradient shrinking K\"ahler-Ricci soliton as an analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a Euclidean space. We also prove an analog of a result of H.-D. Cao and H. Li about the non-existence of compact self-expanders in a Euclidean space.

In this work we show that the concept of a one-parameter persistent rigid-body motion is a slight generalisation of a class of motions called Ribaucour motions by Study. This allows a simple description of these motions in terms of their axode surfaces. We then investigate other special rigid-body motions, and ask if these can be persistent. The special motions studied are line-symmetric motions and motions generated by the moving frame adapted to a smooth curve. We are able to find geometric conditions for the special motions to be persistent and, in most cases, we can describe the axode surfaces in some detail. In particular, this work reveals some subtle connections between persistent rigid-body motions and the classical differential geometry of curves and ruled surfaces.

In this paper, we will show that for any link L in a closed orientable 3-manifold M, infinitely many hyperbolic 3-manifolds are obtained from M by Dehn surgeries so that each of them admits an orientation-preserving smooth finite cyclic group action with fixed point set L.

The purpose of this paper is to study $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifolds of a paracosymplectic manifold $\widetilde{M}$. We prove that the distributions associated with the definition of $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold $M$ are always integrable. A necessary and sufficient condition for an isometrically immersed $\mathcal{P}\mathcal{R}$-semi-invariant submanifold of $\widetilde{M}$ to be a $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold is obtained in terms of the shape operator.

Suppose that G is a locally compact group, that \({\Gamma}\) is a discrete, finitely generated group, and that $$\varphi\colon G \longrightarrow \Gamma$$ is an ‘abstract’ surjective homomorphism. We are interested in conditions which imply that \({\varphi}\) is automatically continuous. We obtain a complete answer to this question in the case where G is a topologically finitely generated locally compact abelian group or an almost connected Lie group. In these two cases the well-known structure theory for such groups G leads quickly to a solution. The question becomes much more difficult if one assumes only that G is a locally compact group. This leads to interesting questions about normal subgroups in infinite products and in ultraproducts. Łos’ theorem, the solution of the 5th Hilbert problem, and recent results by Nikolov–Segal can be combined to answer the question.