In this paper the author finds and describes all similarity space motions, which have only plane trajectories of points. All such motions are explicitly expressed. They are of 5 types, all of them cylindrical. Trajectories are conic sections (3 types) or arbitrary plane curves (2 types).

Let ${\Sigma }^2$ be an immersed surface in $M^2\left(c\right)\times ℝ$ with constant mean curvature. We consider the traceless Weingarten operator $\phi$ associated to the second fundamental form of the surface, and we introduce a tensor $S$, related to the Abresch-Rosenberg quadratic differential form. We establish equations of Simons type for both $\phi$ and $S$. By using these equations, we characterize some immersions for which $\left|\phi \right|$ or $\left|S\right|$ is appropriately bounded.

We give necessary and sufficient local conditions for the simultaneous unitarizability of a set of analytic matrix maps from an analytic 1-manifold into ${\mathrm{SL}}_nℂ$ under conjugation by a single analytic matrix map.We apply this result to the monodromy arising from an integrable partial differential equation to construct a family of $k$-noids, genus-zero constant mean curvature surfaces with three or more ends in euclidean, spherical and hyperbolic $3$-spaces.

In this paper, geometric properties of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space are investigated by applying the singularity theory of smooth functions from the contact viewpoint.

The objective of this paper is to study singularities of n-ruled (n + 1)-manifolds in Euclidean space. They are one-parameter families of n-dimensional affine subspaces in Euclidean space. After defining a non-degenerate n-ruled (n + 1)-manifold we will give a necessary and sufficient condition for such a map germ to be right-left equivalent to the cross cap × interval. The behavior of a generic n-ruled (n + 1)-manifold is also discussed.

We study Legendre and slant curves for Bianchi-Cartan-Vranceanu metrics. These curves are characterized through the scalar product between the normal at the curve and the vertical vector field and in the helix case they have a proper (non-harmonic) mean curvature vector field. The general expression of the curvature and torsion of these curves and the associated Lancret invariant (for the slant case) are computed as well as the corresponding variant for some particular cases. The slant (particularly...

We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.

The remarkable development of the theory of smooth quasigroups is surveyed.

Define for a smooth compact hypersurface $M^n$ of ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left(M^n\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}(M^n)/r(M^n)$, where $r\left(M^n\right)$ is the distance from $M^n$ to its central set. (In other words, $r\left(M^n\right)$ is the maximal radius of an open non-selfintersecting tube around $M^n$ in ${\mathbf{R}}^{n+1}.)$We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le exp\left(c\right(n\left)d^{\alpha \left(n\right)d^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ depend only on $n$, and rigid isotopy means an isotopy passing only through hypersurfaces of degree...

We introduce a skeletal structure $(M,U)$ in ${ℝ}^{n+1}$, which is an $n$- dimensional Whitney stratified set $M$ on which is defined a multivalued “radial vector field” $U$. This is an extension of notion of the Blum medial axis of a region in ${ℝ}^{n+1}$ with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” $ℬ$. We introduce geometric invariants of the radial vector field $U$ on $M$ and a “radial flow” from $M$ to $ℬ$. Together these allow us to provide sufficient numerical conditions for...