Hiroshi Tamaru

Bio: Hiroshi Tamaru is an academic researcher from Hiroshima University. The author has contributed to research in topics: Lie group & Symmetric space. The author has an hindex of 14, co-authored 55 publications receiving 665 citations. Previous affiliations of Hiroshi Tamaru include Sophia University & Osaka City University.

Cohomogeneity one actions on noncompact symmetric spaces of rank one

Abstract: 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 CH", n ≥ 3. For the quaternionic hyperbolic spaces MH n , n > 3, 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.

Cohomogeneity one actions on noncompact symmetric spaces of rank one

Abstract: 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.

Homogeneous Codimension One Foliations on Noncompact Symmetric Spaces

Abstract: We determine the isometric congruence classes of homogeneous Riemannian foliations of codimension one on connected irreducible Riemannian symmetric spaces of noncompact type. As an application we show that on each connected irreducible Riemannian symmetric space of noncompact type and rank greater than two there exist noncongruent homogeneous isoparametric systems with the same principal curvatures, counted with multiplicities.

Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit

Abstract: We classify all totally geodesic submanifolds of connected irreducible Riemannian symmetric spaces of noncompact type which arise as a singular orbit of a cohomogeneity one action on the symmetric space.

Elementary differential geometry

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Riemannian flag manifolds with homogeneous geodesics

Abstract: A geodesic in a Riemannian homogeneous manifold (M = G/K, g) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group G. We investigate G-invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group G. We use an important invariant of a flag manifold M = G/K, its T-root system, to give a simple necessary condition that M admits a non-standard G-invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R 2l+2 ) = SO(2l +1)/U(l) of complex structures in R 2l+2 , and the complex projective space CP 2l-1 = Sp(l)/U(1) " Sp(l- 1) admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only G-invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra g of G). According to F. Podesta and G.Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.

Cohomogeneity one actions on noncompact symmetric spaces of rank one

Abstract: 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 CH", n ≥ 3. For the quaternionic hyperbolic spaces MH n , n > 3, 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.

On δ-homogeneous Riemannian manifolds

Abstract: We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut δ -homogeneous spaces in the case of Riemannian manifolds and prove that they constitute a new proper subclass of geodesic orbit (g.o.) spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian spaces.

Cohomogeneity one actions on noncompact symmetric spaces of rank one

Abstract: 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.