This note is an extension of the paper [HMP]. The main theorem of [HMP] states that a relatively smooth hypersurface (i.e. a hypersurface whose singular locus has sufficient large codimension with respect to its degree) is unirational. A mild modification can generalize this to complete intersections. As an application, we will show the Fano variety of a relatively smooth hypersurface is also u...

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...

A hypersurface of order (n+ 1)(ph − 1) in projective space of dimension n of prime characteristic p has an invariant monomial. This implies that a hypersurface of order (n+1)(ph−1)−1 determines an invariant point. A hypersurface of order d < n+ 1 in a projective space of dimension n of characteristic two has an invariant set of subspaces of dimension d− 1 determined by one linear condition on t...

In this article, we prove that there exists at least one chord orbit of Hamilton vector field connecting the Legendre submanifold on a given energy stable hypersurface in some symplectic manifolds. Démonstration de existence de chord orbital de hamiltonien en une stable hypersurface dans une variété symplectique. Résumé. Soit Σ une hypersurface close en variété symplectique, Il est de montrer q...

The theories of gravity which violate local Lorentz invariance do not admit a universal maximum speed of signal-propagation. Different field excitations see a different effective metric and hence a different light cone. In these theories, although one can define the Killing horizon in a conventional way, this definition does not capture the notion of a black hole. This is so because there exist...

We prove new estimates for the volume of a Lorentzian manifold and show especially that cosmological spacetimes with crushing singularities have finite volume. 0. Introduction Let N be a (n + 1)-dimensional Lorentzian manifold and suppose that N can be decomposed in the form (0.1) N = N0 ∪N− ∪N+, where N0 has finite volume and N− resp. N+ represent the critical past resp. future Cauchy developm...

In this article, we prove that there exists at least one periodic orbit of Hamilton vector field on a given energy hypersurface in R2n which proves the longstanding Seifert’s conjecture. The more general results are also obtained. Démonstration de la Conjecture de Seifert sur existence de périodique orbital de hamiltonien en une hypersurface dans une variété symplectique. Résumé. Soit Σ une hyp...

Okumura gave a necessary and sufficient condition for an oriented real hypersurface of a Kähler manifold to be a contact metric manifold with respect to the naturally induced almost contact metric structure. In this paper, we discuss an oriented hypersurface of a quasi Kähler manifold and give a necessary and sufficient condition for such a hypersurface to be a quasi contact metric manifold wit...

Let SUX(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P and the map is a double cover branched over a sextic hypersurface called the Coble sextic. In the dual P there is a unique cubic hypersurface, the Coble cubic, singular exactly along the abelian surface of degree 1 line bundles on X. We give a new proof that the...