Jun-ichi Inoguchi

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati–Konno–Ichikawa elastic beam equation, the complex Dym equation and the short pulse equation. They are related to the modified KdV or the sine–Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler–Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.

MINIMAL SURFACES IN 3-DIMENSIONAL SOLVABLE LIE GROUPS

The author studies minimal surfaces in 3-dimensional solvable Lie groups with left invariant Riemannian metrics. A Weierstras type integral representation formula for minimal surfaces is obtained.

PSEUDO-SYMMETRIC CONTACT 3-MANIFOLDS

Contact Homogeneous 3-manifolds are pseudo-symmetric spaces of constant type. All Sasakian 3-manifolds are pseudo-symmetric spaces of constant type.

Biharmonic curves in Minkowski 3-space.

We give a differential geometric interpretation for the classification of biharmonic curves in semi-Euclidean 3-space due to Chen and Ishikawa (1991).

A Weierstrass type representation for minimal surfaces in Sol

The normal Gauss map of a minimal surface in the model space Sol of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.

Darboux transformations on timelike constant mean curvature surfaces

Abstract We give loop group theoretic reformulated Backlund transformations on constant mean curvature timelike surfaces in Minkowski 3-space. Further we present 1-soliton surfaces explicitly.

On time-like surfaces of positive constant Gaussian curvature and imaginary principal curvatures

Abstract We establish the Backlund transformation for the construction of time-like surfaces with positive Gaussian curvature and imaginary principal curvatures. The construction can be realized by algebraic algorithm via Darboux transformations.

Timelike Bonnet surfaces in Lorentzian space forms

Abstract We study timelike surfaces in Lorentzian space forms which admit a one-parameter family of isometric deformations preserving the mean curvature.

Magnetic curves in Sasakian manifolds

In this paper we classify the magnetic trajectories corresponding to contact magnetic fields in Sasakian manifolds of arbitrary dimension. Moreover, when the ambient is a Sasakian space form, we prove that the codimension of the curve may be reduced to 2. This means that the magnetic curve lies on a 3-dimensional Sasakian space form, embedded as a totally geodesic submanifold of the Sasakian space form of dimension (2n+1).

Minimal translation surfaces in the Heisenberg group Nil3

A translation surface in the Heisenberg group Nil3 is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group Nil3.