Nobuko Takeuchi

Generic properties of helices and Bertrand curves

Abstract. We study generic properties of cylindrical helices and Bertrand curves as applications of singularity theory for plane curves and spherical curves.

A closed surface of genus one in ³ cannot contain seven circles through each point

There exists a closed surface of genus one in E3 which contains six cirlces through each point, but any closed surface of genus one in E3 cannot contain seven circles through each point.

Singularities of ruled surfaces in [open face R]3

We study singularities of ruled surfaces in ℝ 3 . The main result asserts that only cross-caps appear as singularities for generic ruled surfaces.

A sphere as a surface which contains many circles

A sphere in E3 is characterized as a compact simply connected C∞ surface which contains three circles of E3 through each point.

A surface which contains planar geodesics

It is proved that a complete surface in E3 is a sphere or a plane if it contains at least four geodesics through each point which are plane curves.

On a system of fifth-order partial differential equations describing surfaces containing two families of circular arcs

Let z = f(x, y) be a germ of a C 5-surface at the origin in ℝ3 containing several continuous families of circular arcs. For examples, we have a usual torus with four such families and R. Blums cyclide with six such families. We introduce the system of fifth-order nonlinear partial differential equations for f which describes such a surface germ completely. As applications, we obtain the analyticity of f, and the finite dimensionality of the solution space of such a system of differential equations. We give a brief survey of [Kataoka K, Takeuchi N. A system of fifth-order partial differential equations describing a surface which contains many circles, UTMS 2012-10 (Preprint series of Graduate School of Mathematical Sciences, the University of Tokyo)] concerning surfaces containing two families of circular arcs.

A sphere can be characterized as a smooth ovaloid which contains one circle through each point

A sphere inE3 can be characterized as a smooth ovaloid which contains one circle of an arbitrary but fixed radius through each point.