Christos Baikoussis

On the Gauss map of ruled surfaces

Let M 2 be a (connected) surface in Euclidean 3-space E 3 , and let G : M 2 →S 2 (1) ⊂ E 3 be its Gauss map. Then, according to a theorem of E. A. Ruh and J. Vilms [3], M 2 is a surface of constant mean curvature if and only if, as a map from M 2 to S 2 (1), G is harmonic, or equivalently, if and only if where δ is the Laplace operator on M 2 corresponding to the induced metric on M 2 from E 3 and where G is seen as a map from M 2 to E 3 . A special case of (1.1) is given by i.e., the case where the Gauss map G:M 2 →E 3 is an eigenfunction of the Laplacian δ on M 2 .

On Legendre curves in contact 3-manifolds

It is first observed that on a 3-dimensional Sasakian manifold the torsion of a Legendre curve is identically equal to +1. It is then shown that, conversely, if a curve on a Sasakian 3-manifold has constant torsion +1 and satisfies the initial conditions at one point for a Legendre curve, it is a Legendre curve. Furthermore, among contact metric structures, this property is characteristic of Sasakian metrics. For the standard contact structure onR3 with its standard Sasakian metric the curvature of a Legendre curve is shown to be twice the curvature of its projection to thexy-plane with respect to the Euclidean metric. Thus this metric onR3 is more natural for the study of Legendre curves than the Euclidean metric.

The Chen-Type of the Spiral Surfaces

We show that a spiral surface M in E3 is of finite type if and only if M is minimal Also, the plane is the only spiral surface in E3 whose the Gauss map G is of finite type, or satisfies the condition ΔG = ΛG, where Λ ∈ R3×3.

Ruled submanifolds with finite type Gauss map

We show that a ruled submanifold with finite type Gauss map in a Euclidean space is a cylinder on a curve of finite type or a plane.

On the Geometry of the 7-Sphere

A sphere of dimension 4n+3 admits three Sasakian structures and it is natural to ask if a submanifold can be an integral submanifold for more than one of the contact structures. In the 7-sphere it is possible to have curves which are Legendre curves for all three contact structures and there are 2 and 3-dimensional submanifolds which are integral submanifolds of two of the contact structures. One of the results here is that if a 3-dimensional submanifold is an integral submanifold of one of the Sasakian structures and invariant with respect to another, it is an integral submanifold of the remaining structure and is a principal circle bundle over a holmophic Legendre curve in complex projective 3-space.

Hypersurfaces of restricted type in Minkowski space

AbstractA submanifold Mnr of Minkowski space

Ruled Surfaces and Tubes with Finite Type Gauss Map

\mathbb{E}_1^m

On the inner curvature of the second fundamental form of helicoidal surfaces

is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of