The object of the present paper is to study the totally umbilical hypersurfaces of weakly projective symmetric spaces and it is shown that the totally umbilical hypersurfaces of weakly projective symmetric space is also a weakly projective symmetric space.

We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergencetype on hypersurfaces of the Euclidean space. We deduce some applications to r-stability as well as to almost-Einstein hypersurfaces.

We enumerate plane algebraic curves with one singular point of any (prescribed) singularity type. It is shown how to generalize the method to the singular hypersurfaces and some cases of enumeration of singular hypersurfaces are solved.

In this article, we prove new stability results for almost-Einstein hypersurfaces of the Euclidean space, based on previous eigenvalue pinching results. Then, we deduce some comparable results for almost umbilical hypersurfaces.

Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space CH with any specified value of the Hopf principal curvature α less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces in terms of Weierstrass-type data, and also obtain a classification of pseudo...

We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms.

Four constructions of constant mean curvature (CMC) hypersurfaces in S are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all values of the mean curvature. Second, ...

In this paper we propose several equivalent definitions of digital curves and hypersurfaces in arbitrary dimension. The definitions involve properties such as one-dimensionality of curves and (n − 1)dimensionality of hypersurfaces that make them discrete analogs of corresponding notions in topology. Thus this work appears to be the first one on digital manifolds where the definitions involve th...

Levi nondegenerate real-analytic hypersurfaces M ⊂ ƒ are well understood due to the seminal results of Tanaka [17] and Chern-Moser[5], where a complete set of local invariants for M has been defined. At the other extreme are those hypersurfaces which are locally CR-equivalent to a product M ׃, (take the real hyperplanes inƒ as a simple example). In-between there is a huge and much less underst...

We describe a strategy for computing Yukawa couplings and the mirror map, based on the Picard-Fuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes [5] in the case of quintic hypersurfaces.) We then explain a technique of Griffiths [14] which can be used to compute the Picard-Fuchs equations of hypersurfaces. Finally, we carry out the computati...