A mandala of Legendrian dualities for pseudo-spheres in semi-Euclidean space

On the Quaternionic Curves in the Semi-Euclidean Space E_4_2

In this study, we investigate the semi-real quaternionic curves in the semi-Euclidean space E_4_2. Firstly, we introduce algebraic properties of semi-real quaternions. Then, we give some characterizations of semi-real quaternionic involute-evolute curves in the semi-Euclidean space E42 . Finally, we give an example illustrated with Mathematica Programme.

Double Point Manifolds of Immersions of Spheres in Euclidean Space

Anyone who has been intrigued by the relationship between homotopy theory and diierential topology will have been inspired by the work of Bill Browder. This note contains an example of the power of these interconnections. We prove that, in the metastable range, the double point manifold a self-transverse immersion S n # R n+k is either a boundary or bordant to the real projective space RP n?k. ...

Tangent Bundle of the Hypersurfaces in a Euclidean Space

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...

Willmore Lagrangian Spheres in the Complex Euclidean Space C

On radii of spheres determined by subsets of Euclidean space

In this paper we consider the problem that how large the Hausdorff dimension of E ⊂ Rd needs to be to ensure the radii set of (d − 1)-dimensional spheres determined by E has positive Lebesgue measure. We obtain two results. First, by extending a general mechanism for studying Falconer-type problems in [4], we prove that it holds when dimH(E) > d− 1+ 1 d and in R2, the index 3 2 is sharp for thi...