Generalized Weierstrass representations for generic surfaces confor-mally immersed into four-dimensional Euclidean and pseudo-Euclidean spaces of different signatures are presented. Integrable deformations of surfaces in these spaces generated by the Davey-Stewartson hierarchy of integrable equations are proposed. Willmore functional of a surface is invariant under such deformations.

A definition of chirality based on group theory is presented. It is shown to be equivalent to the usual one in the case of Euclidean spaces, and it permits to define chirality in metric spaces which are not Euclidean. 2000 MSC code: 58D19 ; PACS code: 02.20.Bb

Last lecture, we spent a considerable amount of effort defining manifolds. We like manifolds because they are locally Euclidean. So, even though it is hard for us to reason about them globally, we know what to do in small neighborhoods. It turns out that this ability is all we really need. This is rather fortunate, because we suddenly have spaces with more interesting structure than the Euclide...

Conformal transformations of a Euclidean (complex) plane have some kind of completeness (sufficiency) for the solution of many mathematical and physical-mathematical problems formulated on this plane. There is no such completeness in the case of Euclidean, pseudo-Euclidean and polynumber spaces of dimension greater than two. In the present paper we show that using the concepts of analogical geo...

In modern theories of unified physical interactions, e.g. superstring theory and M-theory, spacetime has more than four dimensions. There are many works on multidimensional cosmological models (e.g. Refs. 1)–6) and references therein). Some of them 7), 8), 9), 10), 11) are studied in the framework of the scenario of quantum creation of the universe. 12), 13) In the semi-classical approximation,...

we study umbilic (space-like) submanifolds of pseudo-riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo euclidean space and relate this notion to umbilicity. finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.a pseudo-riemannian submanifold m in (a ps...

In the preceding article [T], we have constructed several examples of slant immersions into complex Euclidean spaces. The purpose of this article is to construct some additional examples to get slant and full immersions, in the sense of slantedness, into complex Euclidean spaces with arbitrary dimension, codimension, and positive slant angle.