arXiv research

Study properties of orbits of Hermann actions without commutability assumptions.

The paper studies mean curvature flows on specific orbits of Hermann actions.

We give a formula to calculate the indices of special (non-totally geodesic) minimal orbits of Hermann actions. Also, we give examples of such minimal orbits of Hermann actions and calculate their indices by using the formula.

We first investigate the geometry of orbits of the isotropy action on a semi-simple pseudo-Riemannian symmetric space by investigating the complexified action. Next we investigate the geometry of the orbits of Hermann type actions on the symmetric spaces. By considering two special Hermann type actions on the symmetric…

The paper studies curvatures and austere properties of orbits in symmetric spaces.

We investigate the submanifold geometry of the orbits of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots. As applications, we get a formula for the volumes o…

We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducibl…

In this paper, we investigate a curvature-adapted and proper complex equifocal submanifold in a symmetric space of non-compact type. The class of these submanifolds contains principal orbits of Hermann type actions as homogeneous examples. In future, the results in this paper will be used to give a submanifold geometri…

We give examples of certain kind of minimal orbits of Hermann actions and discuss whether each of the examples is austere.

The paper proves a Calabi-Yau structure on complexifications of rank two symmetric spaces.

We study hyperpolar actions on reducible symmetric spaces of the compact type. Our main result is that an indecomposable hyperpolar action on a symmetric space of the compact type is orbit equivalent to a Hermann action or of cohomogeneity one.

In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.

First we show that a curvature-adapted proper complex equifocal submanifold is a principal orbit of a Hermann type action under certain condition. Next we show that a proper complex equifocal submanifold is curvature-adapted under certain condition.

In this paper, we assume that all isoparametric submanifolds have flat section. The main purpose of this paper is to prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type admits a reflective focal submanifold and if it of real analy…

In this paper, we prove that full irreducible curvature-adapted isoparametric submanifolds of codimension greater than one in a symmetric space of non-compact type are principal orbits of Hermann actions on the symmetric spaces under certain conditions.

Study on $G_{2}$ actions on symmetric spaces, focusing on orbit properties.

We show that integration over a $G$-manifold $M$ can be reduced to integration over a minimal section $Σ$ with respect to an induced weighted measure and integration over a homogeneous space $G/N$. We relate our formula to integration formulae for polar actions and calculate some weight functions. In case of a compact …

In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…

We study isometric actions on Riemannian symmetric spaces of noncompact type which are induced by reductive algebraic subgroups of the isometry group. We show that for such an action there exists a corresponding isometric action on a dual compact symmetric space, which reflects many properties of the original action. F…

It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-ada…

In this paper, we investigate the mean curvature flows for an equifocal submanifold in a symmetric space of compact type and its focal submanifolds as initial data. It is known that equifocal submanifolds of codimension greater than one in irreducible symmetric spaces of compact type occur as principal orbits of Herman…

Goursat showed that in the presence of an intermediate integral, the problem of solving a second-order Monge-Ampere equation can be reduced to solving a first-order equation, in the sense that the generic solution of the first-order equation will also be a solution of the original equation. An attempt by Hermann to giv…

Lectures on polar actions and their properties in Riemannian geometry.

We suggest several mathematical counterparts to the idea of "effective degrees of freedom" and formulate specific questions, much of which are inspired by Larry Guth's results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in bal…

Mathematicians decode geometric properties from eigenvalues over 112 years.

This paper poses some basic questions about instances (hard to find) of a special problem in 3-manifold topology. "Important though the general concepts and propositions may be with the modern industrious passion for axiomatizing and generalizing has presented us...nevertheless I am convinced that the special problems …

A singular (or Hermann) foliation on a smooth manifold $M$ can be seen as a subsheaf of the sheaf $\mathfrak{X}$ of vector fields on $M$. We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie brack…

We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank …

Hermann Schwarz, while studying complex analysis, introduced the geometric interpretation for the Poisson kernel in 1890. We shall see here that the geometric interpretation can be useful to develop a new approach to some old classical problems as well as to obtain several new results, mostly related to hyperbolic geom…

This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the projective closure of the affine plane over the octonion algebra. The final result in th…

Weyl and Cartan proposed different but related ways to handle infinitesimal geometry in the early 1920s.

The paper connects calculus, gauge theory, and noncommutative worlds.

This paper poses some basic questions about instances (hard to find) of a special problem in 3-manifold topology. "Important though the general concepts and propositions may be with the modern industrious passion for axiomatizing and generalizing has presented us ... nevertheless I am convinced that the special problem…

The generalization of Frobenius' theorem to foliations with singularities is usually attributed to Stefan and Sussmann, for their simultaneous discovery around 1973. However, their result is often referred to without caring much on the precise statement, as some sort of magic spell. This may be explained by the fact th…

The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications or mathematical physics. The main purpose of this paper, an extended version of a lecture at the second workshop on Differential Equations by Al…

F.: Good morning Hermann, I would like to talk with you about infinitesimals. G.: Tell me Pierre. F.: I'm fed up of all these slanders about my attitude to be non rigorous, so I've started to study nonstandard analysis (NSA) and synthetic differential geometry (SDG). G.: Yes, I've read something ... F.: Ok, no problem …

We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to stu…

Reduces proper actions to simpler core actions for analysis.

Introduces Conditional Action Trees to simplify RL action spaces.

One problem in the application of reinforcement learning to real-world problems is the curse of dimensionality on the action space. Macro actions, a sequence of primitive actions, have been studied to diminish the dimensionality of the action space with regard to the time axis. However, previous studies relied on human…

Study polar actions on Damek-Ricci spaces, proving existence and finding examples.

Reduction principles for proper actions on smooth manifolds.

Totally geodesic sections found in polar actions.

Simplifies large action space bandits by selecting representative actions.

Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.

Conditions for reducing quasi-actions to tree actions and group properties.

We identify action representations from video data, proving their statistical benefits.

We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.