arXiv subjects

Masahiro Morimoto

Publications and source records attributed to Masahiro Morimoto.

8 recordsLinked to original sources

arXiv2025

The parallel transport map over reductive homogeneous space

We show that the parallel transport map over a reductive homogeneous space with natural torsion-free connection becomes an affine submersion with horizontal distribution. This generalizes one of the main results in the author's previous paper in the case of affine symmetric spaces. We also prove the compactness of the shape operators of the submanifold lifted by the parallel transport map. This improves a previous result by the author and generalizes some results of Terng-Thorbergsson and of Koike. Furthermore we propose two definitions for the regularized mean curvatures of affine Fredholm submanifolds in Hilbertable spaces and discuss their relations to the parallel transport map. In particular, each fiber of the parallel transport map over a reductive homogeneous space is shown to be minimal in both senses.

math.DG↗

arXiv2023

Non-existence of exceptional orbits under polar actions on Hilbert spaces

We prove that any polar action on a separable Hilbert space by a connected Hilbert Lie group does not have exceptional orbits. This generalizes a result of Berndt, Console and Olmos in the finite dimensional Euclidean case. As an application, we give a simple geometric proof of the fact that any hyperpolar action on a simply connected compact Riemannian symmetric space by a connected Lie group does not have exceptional orbits.

math.DG↗

arXiv2023

Path group actions induced by sigma-actions and affine Kac-Moody symmetric spaces of group type

In 1995, C.-L. Terng associated to each hyperpolar action on a compact symmetric space, a hyperpolar proper Fredholm (PF) action on a Hilbert space. This is a group action by an infinite dimensional path group and it acts on a Hilbert space via the gauge transformations. Those two hyperpolar actions are related through an equivariant Riemannian submersion called the parallel transport map and they have close relations to the infinite dimensional symmetric spaces called affine Kac-Moody symmetric spaces. In this paper we define a linear isomorphism between Hilbert spaces and show that it is equivariant with respect to the gauge transformations and is compatible with the parallel transport map. Using this isomorphism we extend and unify all known computational results of principal curvatures of PF submanifolds in Hilbert spaces. Especially we study the submanifold geometry of orbits of hyperpolar PF actions associated to sigma-actions and give new examples of austere PF submanifolds in Hilbert spaces. Moreover we show that the isomorphism between Hilbert spaces given here corresponds to a natural isomorphism between affine Kac-Moody symmetric spaces of group type.

math.DG↗

arXiv2022

Curvatures and austere property of orbits of path group actions induced by Hermann actions

It is known that an isometric action of a Lie group on a compact symmetric space gives rise to a proper Fredholm action of a path group on a path space via the gauge transformations. In this paper, supposing that the isometric action is a Hermann action (i.e. an isometric action of a symmetric subgroup of the isometry group) we give an explicit formula for the principal curvatures of orbits of the path group action and study the condition for those orbits to be austere, that is, the set of principal curvatures in the direction of each normal vector is invariant under the multiplication by minus one. To prove the results we essentially use the facts that Hermann actions are hyperpolar and all orbits of Hermann actions are curvature-adapted submanifolds. The results greatly extend the author's previous result in the case of the standard sphere and show that there exist a larger number of infinite dimensional austere submanifolds in Hilbert spaces.

math.DG↗

arXiv2020

On weakly reflective submanifolds in compact isotropy irreducible Riemannian homogeneous spaces

We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.

math.DG↗

arXiv2019

On weakly reflective PF submanifolds in Hilbert spaces

A weakly reflective submanifold is a minimal submanifold of a Riemannian manifold which has a certain symmetry at each point. In this paper we introduce this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist so many infinite dimensional weakly reflective PF submanifolds in Hilbert spaces. In particular each fiber of the parallel transport map is shown to be weakly reflective. These imply that in infinite dimensional Hilbert spaces there exist so many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.

math.DG↗

arXiv2019

Austere and arid properties for PF submanifolds in Hilbert spaces

Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce these two notions into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds in Hilbert spaces. We also mention a classification problem of minimal orbits in hyperpolar PF actions on Hilbert spaces.

math.DG↗

arXiv2017

The Heat Operator of a Transversally Elliptic Operator

Let $G$ be a connected compact Lie group. We study the heat operator of a $G$-transversally elliptic operator. After we review the spectral properties of a $G$-transversally elliptic operator, we define the character, that is a distribution on $G$ generalizing the trace of the heat operator to the $G$-equivariant case. The main theorem of this paper gives the estimate of $f_α(t)$, which essentially determines the convergence of the character.

math.DG↗