Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we study the geometry of a semi-Riemannian manifold ¯ M of quasi-constant curvature. The main result is two characterization theorems for ¯ M admits solenoidal and screen totally umbilical r (> 1)-lightlike subm...

For an n-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type called W (n,F)-Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the W (n,F)-Willmore hypersurface, for which the variational equation and Simons' type integral equalities are obtained. Moreover, we construct a few examples of W (n,F)-Willmore hy...

In this paper we introduce the concept of a bipolar variety of a real algebraic hypersurface. This notion is then used for the design and complexity estimations of a novel type of algorithms that finds algebraic sample point for the connected component of a singular real hypersurface. The complexity of these algorithms is polynomial in the maximal geometric degree of the bipolar varieties of th...

We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov [16] for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic hypersurface singularities.

CONTEXT Isolated renal cell carcinoma recurrence at the renal fossa is a rare event. This condition occurs in 1 to 2% of radical nephrectomy cases. It is usually seen in postoperative follow-up imaging examinations such as abdominal computed tomography or abdominal ultrasound. There is controversy among urologists and oncologists regarding the best way to treat this rare situation, because of t...

We study the forms of curvatures of lightlike hypersurfaces M of an indefinite Kenmotsu manifold M̄ subject to the conditions: (1) M is locally symmetric, i.e., the curvature tensor R of M be parallel on TM , or (2) M is a semi-symmetric manifold, i.e., R(X, Y )R = 0 on TM . M.S.C. 2010: 53C25, 53C40, 53C50.

Introduction. In my note on A 5 curve theorem generalizing the theorem of Carnot, I introduced the notion of curve cross ratio. This extension of the ordinary cross ratio is the simplest situationally invariant (3.4) case of the generalized or hypersurface cross ratio of n + l pairs of hypersurfaces in w-space which is the subject of the following lines. The generalized cross ratio is a t the s...