Naoyuki Koike

Actions of Hermann type and proper complex equifocal submanifolds

In this paper, we mainly prove that principal orbits of an acti on of Hermann type on a symmetric space of non-compact type are curvature a dapted and proper complex equifocal. The proof is performed by showing that pri ncipal orbits of the action are partial tubes over a totally geodesic singular or bit and investingating the shape operatores of the partial tubes over a submanifold in a symmetric space of non-compact type.

Examples of a complex hyperpolar action without singular orbit

La nocion de una accion hiperpolar compleja sobre un espacio simetrico de tipo no compacto fue recientemente introducida como el analogo de la accion hiperpolar sobre un espacio simetrico de tipo compacto. Como ejemplos de una accion hiperpolar complejas, nosotros tenemos acciones de tipo Hermann, las cuales admiten una orbita (o un punto fijo) singular totalmente geodesica excepto para un ejemplo. Todas las orbitas principales de acciones de tipo Hermann son curvatura-adaptadas y unifocales complejas propias. este articulo, nosotros damos algunos ejemplos de una accion hiperpolar compleja sin orbitas singulares como grupo soluble de acciones libres y encontramos acciones complejas hiperpolares cuyas orbitas son no curvatura-adaptadas o no propias unifocales complejas. Tambien, mostramos que algunos de los ejemplos poseen solamente orbitas minimales.

On Curvature-Adapted and Proper Complex Equifocal Sub- manifolds

In this paper, we investigate curvature-adapted and proper complex equifocal submanifolds in a symmetric space of non-compact type. The class of these submani- folds contains principal orbits of Hermann type actions as homogeneous examples and is included by that of curvature-adapted and isoparametric submanifolds with flat section. First we introduce the notion of a focal point of non-Euclidean type on the ideal bound- ary for a submanifold in a Hadamard manifold and give the equivalent condition for a curvature-adapted and complex equifocal submanifold to be proper complex equifocal in terms of this notion. Next we show that the complex Coxeter group associated with a curvature-adapted and proper complex equifocal submanifold is the same type group as one associated with a principal orbit of a Hermann type action and evaluate from above the number of distinct principal curvatures of the submanifold.

The Decomposition of Curvature Netted Hypersurfaces

We define the concept of a curvature netted hypersurface and investigate in what case the hypersurface is covered by a twisted product of spheres (or topological product of spheres). All hypersurfaces in a space form such that the number of mutually distinct principal curvatures is constant (i.e. each principal curvature has constant multiplicity) are curvature netted hypersurfaces. Also, we state some inductive constructions of the hypersurfaces, where we use the discussion related to the tube.

Equiaffine immersions of general codimension and its transversal volume element map

With an equiaffine immersion of codimension 1 into the affine space with the natural equiaffine structure, the conormal map is associated. In this paper, for an equiaffine immersion of general codimension into the space, we shall define the map corresponding to the conormal map, which is called the transversal volume element map. And we shall investigate if, an equiaffine immersion of general codimension into the space is determined by its affine fundamental form and its transversal volume element map.

The Lipschitz-Killing curvature for an equiaffine immersion and theorems of Gauss-Bonnet type and Chern-Lashof type

In this paper, we define an equiaffine immersion of general codimension and the Lipschitz-Killing curvature for the immersion. Furthermore, we prove theorems of Gauss-Bonnet type and Chern-Lashof type for the immersion.

An extrinsic decomposition theorem and a slant tube theorem for a curvature netted hypersurface

In this paper, we treat hypersurfaces in a Euclidean space the number of whose distinct principal curvatures is constant almost everywhere. We call such a hypersurface satisfying certain additional condition a curvature netted hypersurface. First we shall define the notions of a twisted (or warped) sum immersion, a slant focal map and a slant tube. We shall investigate, in what case, a complete curvature netted hypersurface is immersed by a warped sum immersion or becomes a slant tube of the image of a slant focal map.

A modfied mean curvature flow in Euclidean space and soap bubbles in symmetric spaces

In this paper, we show that small spherical soap bubbles in irreducible simply connected symmetric spaces of rank greater than one are constructed from the limits of a certain kind of modified mean curvature flows starting from small spheres in the Euclidean space of dimension equal to the rank of the symmetric space, where we note that the small spherical soap bubbles are invariant under the isotropy subgroup action of the isometry group of the symmetric space. Furthermore, we investigate the shape and the mean curvature of the small spherical soap bubbles.

Volume-Preserving Mean Curvature Flow for Tubes in Rank One Symmetric Spaces of Non-compact Type

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the tubeness is preserved along the flow in the case where the ambient space is a rank one symmetric space of non-compact type, the reflective submanifold is an invariant submanifold and the radius function of the initial tube is radial. Furthermore, in this case, we prove that the flow reaches to the invariant submanifold or it exists in infinite time and converges to another tube of constant mean curvature in the