Hypersurfaces With Pointwise 1-Type Gauss Map

Abstract

In this paper we prove that an oriented hypersurface M of a Euclidean space E n+1 has pointwise 1-type Gauss map of the first kind if and only if M has constant mean curvature. Then we conclude that all oriented isoparametric hypersurfaces of E n+1 has 1-type Gauss map. We also show that a rational hypersurface of revolution in a Euclidean space E n+1 has pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.

On Spacelike Rotational Surfaces With Pointwise 1-Type Gauss Map

“…In this work, we study double spacelike rotational surfaces defined by (10) in E 4 1 whose meridians lie in spacelike 2-planes. By choosing β(s) = (x(s), 0, z(s), 0) in the x 1 x 3 -plane, we have from (10) a rotational surface E 4 1 given by (11)…”

“…Otherwise, it is said to be of the second kind (cf. [1,3,5,6,7,10,12,14,17,19]). The complete classification of ruled surfaces in E and e n+1 , e n+2 are normal to M .…”

Shape Operator and Gauss Map of Pointwise 1-Type

“…In particular, if M is a hypersurface of E m , the Gauss map G is obviously identified with a unit normal vector field on M . It is well known that mean curvature is constant if and only if the Gauss map is of pointwise 1-type of the first kind when a submanifold M is a hypersurface of Euclidean space ( [5,12]). …”

“…for some non-zero smooth function f and a constant vector C. Such a Gauss map is called of pointwise 1-type ( [5,7,9,12,13,14,16]). In particular, if C = 0, it is said to be of pointwise 1-type of the first kind.…”

FLAT ROTATIONAL SURFACES WITH POINTWISE 1-TYPE GAUSS MAP IN E4

“…Otherwise, the pointwise 1-type Gauss map is said to be of the second kind. Surfaces in Euclidean space and in pseudo-Euclidean space with pointwise 1-type Gauss map were recently studied in [7], [8], [10], [11], [12], [13], [14]. Also Dursun and Turgay in [9] gave all general rotational surfaces in E 4 with proper pointwise 1-type Gauss map of the first kind and classified minimal rotational surfaces with proper pointwise 1-type Gauss map of the second kind.…”