arXiv research

We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups.

Study extends reflective submanifold theory to compact homogeneous spaces.

In this paper, we give a necessarly and sufficient condition for orbits of linear isotropy representations of Riemannian symmetric spaces are biharmonic submanifolds in hyperspheres in Euclidean spaces. In particular, we obtain examples of biharmonic submanifolds in hyperspheres whose co-dimension is greater than one.

Harvey-Lawson and Anciaux introduced the notion of austere submanifolds in pseudo-Riemannian geometry. We give an equivalent condition for an orbit of the isotropy representations for semisimple pseudo-Riemannian symmetric space to be an austere submanifold in a pseudo-sphere in terms of restricted root system theory w…

Let $M$ be a most singular orbit of the isotropy representation of a simple symmetric space. Let $(ν_i, Φ_i)$ be an irreducible factor of the normal holonomy representation $(ν_pM, Φ(p))$. We prove that there exists a basis of a section $Σ_i\subset ν_i$ of $Φ_i$ such that the corresponding shape operators have rational…

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be reg…

We write down the local equations that characterize the submanifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of TM endowed with the tangent Dirac structure. In the Poisson case, these formulas prove again a result of Xu: the submanifold N has a normal …

Study circle actions on manifolds with 3 fixed points, finding dimension constraints and unique structures.

We prove a Berger type theorem for the normal holonomy group (i.e., the holonomy group of the normal connection) of a full complete complex submanifold of the complex projective space. Namely, if the normal holonomy does not act transitively, then the submanifold is the complex orbit, in the complex projective space, o…

Classifies geodesic orbit spaces with abelian isotropy subgroups.

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…

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

Classifies multigraphs for torus actions on 6D manifolds with isolated fixed points.

Study of spacelike submanifolds with umbilical lightlike normals in Lorentzian spacetimes.

The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.

Paper studies curvature in Finsler geometry, proving curvature constancy under isotropy.

The paper introduces isotropy as a regularizer to enhance portfolio stability.

We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podesta and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose…

A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. Such group actions have been studied extensively on symmetric spaces. We show how to construct a manifold admitting a polar group action by prescribing their isotropy groups along a fundamental d…

Real flag manifolds are the isotropy orbits of noncompact symmetric spaces $G/K$. Any such manifold $M$ enjoys two very peculiar geometric properties: It carries a transitive action of the (noncompact) Lie group $G$, and it is embedded in euclidean space as a taut submanifold. The aim of the paper is to link these two …

Study isotropy groups for complex orthogonal and skew-symmetric matrices.

Study of families of lines on spheres and their focal sets.

Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.

Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.

Study the smallest Laplace eigenvalue in special geometric spaces.

Geodesic orbit metrics proven on specific homogeneous spaces.

We classify all simply connected Riemannian manifolds whose isotropy groups act with cohomogeneity less than or equal to two.

Given a singular foliation, we attach an "essential isotropy" group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with t…

The space of $G$-invariant metrics on a homogeneous space $G/H$ is in one-to-one correspondence with the set of inner products on the tangent space $\fr{m}\cong T_{\it o}(G/H)$, which are invariant under the isotropy representation. When all the isotropy summands are inequivalent to each other, then the metric is calle…

Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type $D_m$. Based on this geometric interpretation he conjectured that these polynomials…

It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let M^n, n>=2, be a full and irreducible homogeneous submanifold of the sphere $S^{N-1}\subset R^N$ and such that the normal holonomy group is not transitive (on t…

The concept of an objective spatial direction in special relativity is investigated and theories assuming light-speed isotropy while accepting the existence of a privileged spatial direction are classified. A natural generalization of the proper time principle is introduced which makes it possible to devise experimenta…

We classify the rational differential 1-forms with simple poles and simple zeros on the Riemann sphere according to their isotropy group; when the 1-form has exactly two poles the isotropy group is isomorphic to $\mathbb{C}^{*}$, namely $\{z\mapsto az\ \vert\ a\in\mathbb{C}, a\neq0\}$, and when the 1-form has $k\geq 3$…

The paper classifies geodesic orbit spaces with simple isotropy groups.

Study equigeodesics on compact homogeneous spaces using Lie algebra properties.

We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…

We extend the equivariant classification results of Escher and Searle for closed, simply connected, non-negatively curved Riemannian $n$-manifolds admitting isometric isotropy-maximal torus actions to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In particular, we prove…

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic…

We classify the $5$-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 2 of 3) classifies those in which the linear isotropy representation is either irreducible or trivial. The $5$-dimensional geometries with irreducible isotropy are the irreducible Riemannian symmetric spaces, while …

The isotropy action on certain symmetric spaces is shown to be equivariantly formal.

Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.

Calculates affine transformations for specific homogeneous spaces.

New method predicts nonfactuality in LLM responses using semantic isotropy.

We classify homogeneous pseudo-Riemannian manifolds of index 4 which admit an invariant almost hyper-Hermitian structure and an H-irreducible isotropy group. The main result is that all these spaces are flat except in dimension 12.

We obtain an algorithmic construction of the isotropy lattice for a lifted action of a Lie group $G$ on $TM$ and $T^*M$ based only on the knowledge of $G$ and its action on $M$. Some applications to symplectic geometry are also shown.

We use the equivariant Yang-Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4-manifolds.

We give a Riemannian structure to the set $Σ$ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of $Σ$ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold $Σ$ is a universal model for symmetric spaces of the nonc…

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homo…