Dong-Soo Kim

HELICOIDAL SURFACES WITH POINTWISE 1-TYPE GAUSS MAP

The helicoidal surfaces with pointwise 1-type or harmonic gauss map in Euclidean 3-space are studied. The notion of pointwise 1- type Gauss map is a generalization of usual sense of 1-type Gauss map. In particular, we prove that an ordinary helicoid is the only genuine he- licoidal surface of polynomial kind with pointwise 1-type Gauss map of the flrst kind and a right cone is the only rational helicoidal surface with pointwise 1-type Gauss map of the second kind. Also, we give a charac- terization of rational helicoidal surface with harmonic or pointwise 1-type Gauss map.

A Characterization of Ellipses

Remark. Our proof shows that at corner points, the Koch curve doesn’t even have one-sided tangent lines. (Depending on the initial choice of the point x ′ n , the foregoing discussion establishes the nonexistence of a tangent line approaching a corner point “from the left” or “from the right.”) A refined variant of our argument can be used to show that the same is true also for the noncorner points. This implies that no parametrization of the Koch curve can have at any point a nonzero leftor right-hand derivative.

Some Characterizations of Parabolas

We study some properties of tangent lines of parabolas. As a result, we establish some characterizations of parabolas.

CONFORMAL VECTOR FIELDS AND TOTALLY UMBILIC HYPERSURFACES

In this article, we show that if a semi-Riemannian space form carries a conformal vector field V of which the tangential part on a connected hypersurface ecomes a conformal vector field and the normal part on does not vanish identically, then is totally umbilic. Furthermore, we give a complete description of conformal vector fields on semi-Riemannian space forms.

Center of Gravity and a Characterization of Parabolas

Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord

SPACES OF CONFORMAL VECTOR FIELDS ON PSEUDO-RIEMANNIAN MANIFOLDS

AB

SHAPE OPERATOR AH FOR SLANT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS

on the parabola, let us denote by

ON RULED SURFACES IN MINKOWSKl SPACES

P

Some characterizations of spheres and elliptic paraboloids II

the point on the parabola where the tangent is parallel to

FINITE TYPE RULED SURFACES IN LORENTZ-MINKOWSKI SPACE

AB