COMPLETE HYPERSURFACES WITH CONSTANT SCALAR CURVATURE IN A SPECIAL KIND OF LOCALLY SYMMETRIC MANIFOLD

In this paper, we investigate n-dimensional complete and orientable hypersufaces M n (n ≥ 3) with constant normalized scalar curvature in a locally symmetric manifold. Two rigidity theorems are obtained for these hypersurfaces.     

Existence of global invariant jet differentials on projective hypersurfaces of high degree

Let be a smooth complex projective hypersurface. In this paper we show that, if the degree of X is large enough, then there exist global sections of the bundle of invariant jet differentials of order n on X, vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the log-pair , where D is a smooth irreducible divisor of high degree. Moreover, these result are sharp, i.e. one cannot have such jet differentials of order less than n.     

Halphen pencils on weighted Fano threefold hypersurfaces

On a general quasismooth well-formed weighted hypersurface of degree Σ _i=1⁴ a _i in ℙ(1, a ₁, a ₂, a ₃, a ₄), we classify all pencils whose general members are surfaces of Kodaira dimension zero.      

Geometric Potential and Dirac Quantization

A fundamental problem regarding the Dirac quantization of a free particle on an () curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature‐induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin‐layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator‐ordering‐free section is identified and is then found sufficient to lead to the expected form of geometric potential.     

Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces

In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σ ⁿ immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space , and whose boundary is contained into a slice {t} × M ⁿ . By considering the hyperbolic caps of the Lorentz–Minkowski space , we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces immersed with some positive constant r-mean curvature into a Lorentzian product space. For instance, when the ambient space–time is spatially closed, we show that such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition for their existence.     

On quasi-umbilical locally strongly convex homogeneous affine hypersurfaces

In this paper, by developing the techniques of F. Dillen and L. Vrancken in [6], we study quasi-umbilical locally strongly convex homogeneous unimodular-affine hypersurfaces. We will present a characterization of a certain subclass in all dimensions; finally, in dimension five, we will give a complete classification of all quasi-umbilical homogeneous unimodular-affine hypersurfaces.     

Classifications of Isoparametric Hypersurfaces in Randers Space Forms

In this paper, we give the complete classifications of isoparametric hypersurfaces in Randers space forms. By studying the principal curvatures of anisotropic submanifolds in a Randers space (N, F) with the navigation data (h, W), we find that a Randers space form (N, F, dμ_BH) and the corresponding Riemannian space (N, h) have the same isoparametric hypersurfaces, but in general, their isoparametric functions are different. We give a necessary and sufficient condition for an isoparametric function of (N, h) to be isoparametric on (N, F, dμ_BH), from which we get some examples of isoparametric functions.     

A model for free-radical reactions

Using an EHT type Hamiltonian the Heitler-London-VB treatment of the three-centre three-electron problem is simplified such that the interesting features of the potential hypersurfaces of arbitrary three-centre three-electron systems can be obtained simply from the knowledge of atomic orbital energies and overlap integrals. The method is applied to the hypersurfaces of the H₃ system and extensions of the discussion of general radical reactions are indicated.