Hypersurfaces in the quaternions Ⅱ

The Möbius Geometry of Hypersurfaces, Ii

The density of rational points on non-singular hypersurfaces, II

For any integers d, n ≥ 2, let X ⊂ P be a non-singular hypersurface of degree d that is defined over Q. The main result in this paper is a proof that the number NX(B) of Q-rational points on X which have height at most B satisfies NX(B) = Od,ε,n(B n−1+ε), for any ε > 0. The implied constant in this estimate depends at most upon d, ε and n. Mathematics Subject Classification (2000): 11D45 (11G35...

On hypersurfaces in a locally affine Riemannian Banach manifold II

In our previous work (2002), we proved that an essential second-order hypersurface in an infinite-dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature. In this note, we prove the converse; in other words, we prove that a hypersurface of constant nonzero Riemannian curvature in a locally affine (flat) semiRiemannian Banach space is an essen...

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...

The Forgotten Quaternions

A quick glance through the Mathematics section of the 2004 Stanford University Bulletin reveals five courses devoted to complex analysis, but no courses even mentioning quaternions. How is it that complex analysis, a subject that has suffered through hundreds of years of skepticism and distrust, came to be so widely accepted today, while quaternionic analysis, a modern subject immediately accep...