Hiroshi Tamaru

Two-point homogeneous quandles with prime cardinality

Quandles can be regarded as generalizations of symmetric spaces. Among symmetric spaces, two-point homogeneous Riemannian manifolds would be the most fundamental ones. In this paper, we define two-point homogeneous quandles analogously, and classify those with prime cardinality.

A sufficient condition for congruency of orbits of Lie groups and some applications

We give a sufficient condition for isometric actions to have the congruency of orbits, that is, all orbits are isometrically congruent to each other. As applications, we give simple and unified proofs for some known congruence results, and also provide new examples of isometric actions on symmetric spaces of noncompact type which have the congruency of orbits.

The local orbit types of symmetric spaces under the actions of the isotropy subgroups

Abstract The local orbit types of the actions of the isotropy subgroups on compact semi-simple symmetric spaces are investigated. All the possible isotropy subalgebras can be obtained from subdiagrams of the extended Dynkin diagram of the restricted root systems.

Three-dimensional solvsolitons and the minimality of the corresponding submanifolds

In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.

Lie groups locally isomorphic to generalized Heisenberg groups

We classify connected Lie groups which are locally isomorphic to generalized Heisenberg groups. For a given generalized Heisenberg group N, there is a one-to-one correspondence between the set of isomorphism classes of connected Lie groups which are locally isomorphic to N and a union of certain quotients of noncompact Riemannian symmetric spaces.

Realizations of some contact metric manifolds as Ricci soliton real hypersurfaces

Abstract Ricci soliton contact metric manifolds with certain nullity conditions have recently been studied by Ghosh and Sharma. Whereas the gradient case is well-understood, they provided a list of candidates for the nongradient case. These candidates can be realized as Lie groups, but one only knows the structures of the underlying Lie algebras, which are hard to be analyzed apart from the three-dimensional case. In this paper, we study these Lie groups with dimension greater than three, and prove that the connected, simply-connected, and complete ones can be realized as homogeneous real hypersurfaces in noncompact real two-plane Grassmannians. These realizations enable us to prove, in a Lie-theoretic way, that all of them are actually Ricci soliton.

The Solvable Models of Noncompact Real Two-Plane Grassmannians and Some Applications

Every Riemannian symmetric space of noncompact type is isometric to some solvable Lie group equipped with a left-invariant Riemannian metric. The corresponding metric solvable Lie algebra is called the solvable model of the symmetric space. In this paper, we give explicit descriptions of the solvable models of noncompact real two-plane Grassmannians, and mention some applications to submanifold geometry, contact geometry, and geometry of left-invariant metrics.

The Space of Left-Invariant Riemannian Metrics

Geometry of left-invariant Riemannian metrics on Lie groups has been studied very actively. We have proposed a new framework for studying this topic from the viewpoint of the space of left-invariant metrics. In this expository paper, we introduce our framework, and mention two results. One is a generalization of Milnor frames, and another is a characterization of solvsolitons of dimension three in terms of submanifold geometry.

On Totally Geodesic Surfaces in Symmetric Spaces of Type AI

We develop an approach to the classification of nonflat totally geodesic surfaces in Riemannian symmetric spaces of noncompact type. In this paper, we concentrate on the case of symmetric spaces of type AI, and show that such surfaces correspond to certain nilpotent matrices. As applications, we obtain explicit classifications in the cases of rank two and three.

Two-Step Nilpotent Lie Groups and Homogeneous Fiber Bundles

In this paper, we construct two-step nilpotent Lie groups from homogeneous fiber bundles over compact symmetric spaces. The structure of the constructed nilpotent groups is expressed in terms of the compact Lie groups involved in the fiber bundles. There are close relations between the geometric properties of the nilpotent groups and the total spaces of the fiber bundles. We will find new examples of nilpotent groups which are weakly symmetric and Riemannian geodesic orbit spaces.