General Rotational Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

Abstract

Abstract. We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classification of minimal general rotational surfaces of elliptic and hyperbolic type, general rotational surfaces with parallel normalized mean curvature vector field, flat general rotational surfaces, and general rotational surfaces with flat normal connection.

Special Structures on General Rotational Surfaces in Pseudo-Euclidean 4-Space With Neutral Metric

“…Lorentz general rotational surfaces in E 4 2 with plane meridian curves are studied in [2] where the complete classification of some special subclasses is given, namely: minimal, flat, with flat normal connection, with parallel normalized mean curvature vector field. A particular subcase is the analogue of the Vranceanu rotational surface in the Euclidean space E 4 which is obtained choosing a = b = 1, x(s) = u(s) cos s, and z(s) = u(s) sin s for some smooth function u(s) [27].…”

“…If both a and b are nonzero constants the surface defined by (3.11) is a general rotational surface of hyperbolic type in E 4 2 . Lorentz general rotational surfaces of hyperbolic type with plane meridian curves are studied in [2].…”

“…The general rotational surfaces in E 4 2 parametrized by (3.3) and (3.4) are of elliptic type (the elliptic cosine and sine functions are used in the parametrization). Similar to the general rotational surfaces of elliptic type, in [2], Aleksieva, Turgay and the second author define general rotational surfaces of hyperbolic type in E 4 2 as follows:…”

“…A metric in the case of four-dimensional manifolds that is neither Riemannian nor Lorentz must be of neutral signature (see [4], [9], for example). General rotational surfaces in pseudo-Euclidean spaces of neutral signature were studied by F. Aksoyak, Y. Yayli [1], as well as by Y. Aleksieva and the second author [2].…”

On Pseudo-Umbilical Rotational Surfaces with Pointwise 1-type Gauss Map in $\mathbb{E}^4_2$

“…Recently, Y. Aleksieva, V. Milousheva and N. C. Turgay studied general rotational surfaces in the pseudo-Euclidean space E 4 2 with zero mean curvature vector in [2] and then the first author, E. Canfes and U. Dursun classified such rotational surfaces with pointwise 1-type Gauss map in [4].…”