Inhomogeneous almost symmetric submanifolds
We completely describe inhomogeneous properly embedded almost symmetric submanifolds of Euclidean space as certain unions of parallel symmetric submanifolds of the ambient Euclidean space.
math.DG↗
arXiv subjects
Claudio Gorodski
Publications and source records attributed to Claudio Gorodski.
Almost symmetric submanifolds
We introduce the class of almost symmetric submanifolds of Euclidean space, a close relative of symmetric submanifolds and (contact) sub-Riemannian symmetric spaces. More specifically, we prove that every full irreducible almost symmetric submanifold of Euclidean space is either: a most singular orbit of an s-representation; or an almost singular orbit, which can be realized as a holonomy tube over a symmetric submanifold; or a codimension 3 submanifold. We also include tables of all examples with Lie-theoretic data. We prove that any inhomogeneous almost symmetric submanifold has cohomogeneity one and describe possible structures, including multiply-warped products. We interpret almost symmetric submanifolds as embeddings of sub-Riemannian symmetric spaces, highlighting the interplay between extrinsic and intrinsic symmetry. We propose the co-index of extrinsic symmetry as new invariant and a potential tool to study and hierarchize highly symmetric submanifolds.
math.DG↗
Topics in polar actions
These are the notes for a series of lectures at the Institute of Geometry and Topology of the University of Stuttgart, Germany, in July 13-15, 2022. We assume basic knowledge of isometric actions on Riemannian manifolds, including the normal slice theorem and the principal orbit type theorem. Lecture 1 introduces polar actions and culminates with Heintze, Liu and Olmos's argument to characterize them in terms of integrability of the distribution of normal spaces to the principal orbits. The other two lectures are devoted to two of Lytchak and Thorbergsson's results. In Lecture 2 we briefly review Riemannian orbifolds from the metric point of view, and explain their characterization of orbifold points in the orbit space of a proper and isometric action in terms of polarity of the slice representation above. In Lecture 3 we present their proof of the fact that variationally complete actions in the sense of Bott and Samelson on non-negatively curved manifolds are hyperpolar. The appendix contains explanations of some results used in the lectures, namely: a more or less self-contained derivation of Wilking's transversal Jacobi equation; a discussion of Cartan's and Hermann's criterions for the existence of totally geodesic submanifolds, and a criterion for the polarity of isometric actions on symmetric spaces.
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Totally geodesic submanifolds and polar actions on Stiefel manifolds
We classify totally geodesic submanifolds of the real Stiefel manifolds of orthogonal two-frames. We also classify polar actions on these Stiefel manifolds, specifically, we prove that the orbits of polar actions are lifts of polar actions on the corresponding Grassmannian. In the case of cohomogeneity-one actions we are able to obtain a classification for all real, complex and quaternionic Stiefel manifolds of $k$-frames.
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Representations of compact simple Lie groups whose orbit space has non-empty boundary
We provide detailed calculations for the classification of representations of compact simple Lie groups with non-empty boundary in the orbit space, first announced in a previous paper [arXiv:2112.00513] by the same authors.
math.DG↗
Actions on positively curved manifolds and boundary in the orbit space
We study isometric actions of compact Lie groups on complete orientable positively curved $n$-manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere by actions of compact simple Lie groups with non-empty boundary. We deduce from this the list of representations of compact simple Lie groups that admit non-trivial reductions. As a tool of special interest, we introduce a new geometric invariant of a compact symmetric space, namely, the minimal number of points in a "spanning set" of the space.
math.DG↗
A moment map for the variety of Jordan algebras
We study the variety of complex $n$-dimensional Jordan algebras using techniques from Geometric Invariant Theory.
math.AG↗
The $κ$-nullity of Riemannian manifolds and their splitting tensors
We consider Riemannian $n$-manifolds $M$ with nontrivial $κ$-nullity "distribution" of the curvature tensor $R$, namely, the variable rank distribution of tangent subspaces to $M$ where $R$ coincides with the curvature tensor of a space of constant curvature $κ$ ($κ\in\mathbb R$) is nontrivial. We obtain classification theorems under diferent additional assumptions, in terms of low nullity/conullity, controlled scalar curvature or existence of quotients of finite volume. We prove new results, but also revisit previous ones.
math.DG↗
Generalized warped products and the $κ$-nullity of Riemannian curvature
In this short survey, we show how two (classes of) known examples of inhomogeneous, curvature homogeneous Riemannian manifolds with nontrivial $κ$-nullity can be seen as deformations of homogeneous metrics along the vertical distribution of an integrable Riemannian submersion. We also pose two open questions.
math.DG↗
A diameter gap for quotients of the unit sphere
We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.
math.MG↗
Representations of low copolarity
We classify irreducible representations of compact connected Lie groups whose orbit space is isometric to the orbit space of a representation of a compact Lie group of dimension~$7$, $8$ or $9$. They turn out to be closely related to symmetric spaces, with one exception only.
math.RT↗
Semisimple symmetric contact spaces
We classify contact manifolds $(M,\mathcal D)$ which are homogeneous under a connected semisimple Lie group $G$, and symmetric in the sense that there exists a contactomorphism of $(M,\mathcal D)$ normalizing $G$, fixing a point $o$ in $M$ and restricting to minus identity along $\mathcal D_o$.
math.DG↗
Highly curved orbit spaces
It is known that the infimum of the sectional curvatures (on the regular part) of orbit spaces of isometric actions on unit spheres in bounded above by $4$. We show that the infimum is $1$ for "most" actions, and determine the cases in which it is bigger than $1$.
math.DG↗
Robust index bounds for minimal hypersurfaces of isoparametric submanifolds and symmetric spaces
We find many examples of compact Riemannian manifolds $(M,g)$ whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the metric $g$ is replaced with $g'$ in a neighbourhood of $g$. Our examples $(M,g)$ consist of certain minimal isoparametric hypersurfaces of spheres; their focal manifolds; the Lie groups $SU(n)$ for $n\leq 17$, and $Sp(n)$ for all $n$; and all quaternionic Grassmannians.
math.DG↗
Focal radii of orbits
We show that every effective action of a compact Lie group $K$ on a unit sphere $S^n$ admits an explicit orbit whose principal curvatures are bounded from above by $4\sqrt{14}$.
math.DG↗
Polar symplectic representations
We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case.
math.RT↗
Representations with $Sp(1)^k$-reductions and quaternion-Kähler symmetric spaces
We classify non-polar irreducible representations of connected compact Lie groups whose orbit space is isometric to that of a representation of a finite extension of $Sp(1)^k$ for some $k>0$. It follows that they are obtained from isotropy representations of certain quaternion-Kähler symmetric spaces by restricting to the "non-$Sp(1)$-factor".
math.DG↗
The curvature of orbit spaces
We investigate orbit spaces of isometric actions on unit spheres and find a universal upper bound for the infimum of their curvatures.
math.DG↗