Masami Sekizawa

On Tangent Sphere Bundles with Small or Large Constant Radius

For a Riemannian manifold M, we determine somecurvature properties of a tangent sphere bundleTrM endowed with the induced Sasaki metric in the case when the constantradius r > 0 of the tangent spheres is either sufficientlysmall or sufficiently large.

Natural transformations of symmetric affine connections on manifolds to metrics on linear frame bundles: a classification

We find all second order natural transformations of symmetric affine connections on manifolds into metrics on linear frame bundles. The complete family of naturally resulting metrics (to a fixed symmetric affine connection) is a real vector space generated by some generalization of two “classical” liftings.

On 3-dimensional Riemannian Σ-spaces

The following result is proved: a 3-dimensional connected and simply connected Riemannian manifold admitting a reduced Σ-structure (in the sense of O. Loos) is either a Riemannian symmetric space or it is isometric to a unimodular Lie group with a left-invariant Riemannian metric. At the same time, we give first nontrivial examples of Riemannian Σ-spaces, which are not “symmetric of finite order”.

Curvatures of the diagonal lift from an affine manifold to the linear frame bundle

We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.

Hypersurfaces of Type Number 2 in the Hyperbolic Four-Space and Their Extensions To Riemannian Geometry

Pseudo-symmetric spaces of constant type in dimension 3 are Riemannian three-manifolds whose Ricci tensor has, at all points, one double eigenvalue and one simple constant eigenvalue. In this paper we give a survey of our published results for the case when the constant Ricci eigenvalue is negative. In particular, we show that three-dimensional hypersurfaces of the hyperbolic space ℍ4 whose second fundamental form has rank 2 belong to this class. An explicit classification is presented in the case when the space admits so-called asymptotic foliation. Based on this, we show some existence theorems about local isometric embeddings of such spaces into ℍ4.