This is the Cayley surface when N = 3. The next few are as follows. x4 = x1x3 + 1 2 x2 2 − x1 x2 + 1 4 x1 4 x5 = x1x4 + x2x3 − x1 x3 − x1x2 2 + x1 x2 − 1 5 x1 5 x6 = x1x5 + x2x4 + 1 2 x3 2 − x1 x4 − 2x1x2x3 − 1 3 x2 3 + x1 x3 + 3 2 x1 x2 2 − x1 x2 + 1 6 x1 . Since the first term in (1) is −xN and this is the only occurrence of this variable, these hypersurfaces are polynomial graphs over the re...

A regular vacuum solution in 5D gravity on the principal bundle with the U (1) structural group is proposed as a 4D wormhole. This solution has two null hypersurfaces where an interchange of the sign of some 5D metric components happens. For a 4D observer living on the base of this principal bundle this is a wormhole with two asymptotically flat Lorentzian (Euclidean) spacetimes connected by a ...

We give, in this paper, a monotonicity formula for the Chern-Moser-Weyl curvature tensor under the action of holomorphic embeddings between Levi non-degenerate hypersurfaces with the same positive signature. As an application, we provide some concrete examples of algebraic Levi non-degenerate hypersurfaces with positive signature that are not embeddable into a hyperquadric of the same signature...

Recently some extensions of the classical Darboux integrability theory to autonomous and non-autonomous polynomial vector fields have been done. The classical Darboux integrability theory and its recent extensions are based on the existence of algebraic invariant hypersurfaces. However the algebraicity of the invariant hypersurfaces is not necessary and the unique necessary condition is the alg...

We extend the framework of general boundary quantum field theory (GBQFT) to achieve a fully local description realistic theories. This requires quantization non-Kähler polarizations which occur generically on timelike hypersurfaces in Lorentzian spacetimes as has been shown recently. this two ways: On one hand we replace Hilbert space states by observables localized hypersurfaces, spirit algebr...

We introduce the epitrochoidal hypersurfaces in four dimensional Euclidean space E^4. serve notations of a geometry Giving definition rotational hypersurface, we define and calculate its differential geometric objects, such as Gauss map curvatures. In end, reveal some relations for curvatures that type hypersurfaces.

In this paper, we study a stability problem of free boundary hypersurfaces, and also capillary ones whose supported on horosphere in hyperbolic space. We prove that umbilical hypersurfaces are only stable immersed horosphere. Using the same method, show totally geodesic hyperplane is type-II hypersurface