arXiv subjects
Publications and source records attributed to Hiroshi Tamaru.
A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity one action without singular orbit. In this paper, we classify Ricci soliton Lie hypersurfaces in the complex hyperbolic spaces.
We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type. We also study several examples where the leaf of symmetry is not of the group type. Interesting examples arise from the unit tangent bundle of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold and the Wallach 24-manifold.
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.
We are studying a relationship between isoparametric hypersurfaces in spheres with four distinct principal curvatures and the moment maps of certain Hamiltonian actions. In this paper, we consider the isoparametric hypersurfaces obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. We prove that the Cartan-Münzner polynomials of these hypersurfaces can be written as squared-norms of the moment maps for some Hamiltonian actions. The proof is based on the structure theory of symmetric spaces.
An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As a consequence, we find many new examples of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type. We apply our conceptual approach to derive explicit classifications of cohomogeneity one actions on some symmetric spaces.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.
In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curvatures of the ambient symmetric spaces. Consequently, all of our solvmanifolds are Einstein, which provide a large number of new examples of noncompact homogeneous Einstein manifolds. We also show that our solvmanifolds are minimal, but not totally geodesic submanifolds of symmetric spaces.
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.
In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the nilradical is of two-step. New examples of Einstein solvmanifolds with three-step and four-step nilradicals are also given.
We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces of dimension greater than two. For the quaternionic hyperbolic spaces of dimension greater than two we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Elie Cartan.