arXiv subjects

Ivan Solonenko

Publications and source records attributed to Ivan Solonenko.

7 recordsLinked to original sources

arXiv2026

Cohomogeneity one actions on symmetric spaces of mixed type

In this article, we study isometric cohomogeneity-one actions on symmetric spaces of mixed type, i.e., those whose universal cover splits as a nontrivial product of symmetric spaces of compact, noncompact, and Euclidean types. We provide a new family of "diagonal" cohomogeneity-one actions on symmetric spaces of the form $\mathbb{R}^n \times M_-$, where $M_-$ is of noncompact type. We show that, with the exception of this family, any cohomogeneity-one action on a symmetric space decomposes as a product of isometric actions on its compact, Euclidean, and noncompact factors. This fully reduces the classification problem for cohomogeneity-one actions to symmetric spaces of a single type.

math.DG↗

arXiv2025

Classification of cohomogeneity-one actions on symmetric spaces of noncompact type

We complete the classification of isometric cohomogeneity-one actions on all symmetric spaces of noncompact type up to orbit equivalence.

math.DG↗

arXiv2022

The spinor and Weierstrass representations of surfaces in space

In this paper, following Sullivan, Kusner, and Schmitt, we study conformal immersions of Riemann surfaces into the three-dimensional Euclidean space. Regarding such immersions as special bundle maps from the tangent bundle of the surface to the cotangent bundle of the 2-dimensional sphere, we generalize the classical Weierstrass representation of minimal surfaces to the case of arbitrary conformal immersions. We study how such an immersion gives rise to a spin structure on the surface together with a pair of spinors and how the immersion itself can be studied by means of these spinors.

math.DG↗

arXiv2022

Automorphisms of real semisimple Lie algebras and their restricted root systems

We prove that every automorphism of the restricted root system of a real semisimple Lie algebra -- when defined properly -- can be lifted to an automorphism of that Lie algebra. In particular, this can be applied to automorphisms of the Dynkin diagram of the restricted root system. We also discuss some applications of this result to the theory of symmetric spaces of noncompact type.

math.DG↗

arXiv2021

Homogeneous codimension-one foliations on reducible symmetric spaces of noncompact type

We extend the classification of homogeneous codimension-one foliations on irreducible Riemannian symmetric spaces of noncompact type obtained by Berndt and Tamaru to the reducible case, thus completing it for all noncompact symmetric spaces.

math.DG↗

arXiv2021

Sections of polar actions

In this short note we provide an elementary proof of the folklore result in the theory of isometric Lie group actions on Riemannian manifolds asserting that sections of polar actions are totally geodesic.

math.DG↗

arXiv2021

Classification of homogeneous hypersurfaces in some noncompact symmetric spaces of rank two

We classify, up to isometric congruence, the homogeneous hypersurfaces in the Riemannian symmetric spaces $\mathrm{SL}(3,\mathbb{H})/\mathrm{Sp}(3), \hspace{1pt} \mathrm{SO}(5,\mathbb{C})/\mathrm{SO}(5),$ and $\mathrm{Gr}^*(2,\mathbb{C}^{n+4}) = \mathrm{SU}(n+2,2)/\mathrm{S}(\mathrm{U}(n+2)\mathrm{U}(2)), \, n \geqslant 1$.

math.DG↗