arXiv subjects

Srdjan Vukmirovic

Publications and source records attributed to Srdjan Vukmirovic.

6 recordsLinked to original sources

arXiv2026

Normal sub-Riemannian geodesics related to filtrations of Lie algebras

There is a natural way to construct sub-Riemannian structures that depend on $n$ parameters on compact Lie groups. These structures are related to the filtrations of Lie subalgebras $\mathfrak g_0 < \mathfrak g_1 < \mathfrak g_2 < \dots < \mathfrak g_{n-1}<\mathfrak g_n=\mathfrak g=Lie(G)$. In the case where $n=1$, the explicit solution for normal sub-Riemannian geodesics was provided by Agrachev, Brockett, and Jurjdevic. We extend their solution to apply to general chains of Lie subgroups. Additionally, we describe normal geodesic lines of the induced sub-Riemannian structures on homogeneous spaces $G/K$, where $\mathfrak g_0=Lie(K)$.

math.DG↗

arXiv2024

Integrable systems associated to the filtrations of Lie algebras

In 1983 Bogoyavlenski conjectured that if the Euler equations on a Lie algebra $\mathfrak g_0$ are integrable, then their certain extensions to semisimple lie algebras $\mathfrak g$ related to the filtrations of Lie algebras $\mathfrak g_0\subset \mathfrak g_1\subset \mathfrak g_2\dots\subset\mathfrak g_{n-1}\subset \mathfrak g_n=\mathfrak g$ are integrable as well. In particular, by taking $\mathfrak g_0=\{0\}$ and natural filtrations of $\mathfrak{so}(n)$ and $\mathfrak{u}(n)$, we have Gel'fand-Cetlin integrable systems. We proved the conjecture for filtrations of compact Lie algebras $\mathfrak g$: the system are integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given.

nlin.SI↗

arXiv2021

On the moduli spaces of left invariant metrics on cotangent bundle of Heisenberg group

The main focus of the paper is the investigation of moduli space of left invariant pseudoRiemannian metrics on the cotangent bundle of Heisenberg group. Consideration of orbits of the automorphism group naturally acting on the space of the left invariant metrics allows us to use the algebraic approach. However, the geometrical tools, such as classification of hyperbolic plane conics, will often be required. For metrics that we obtain in the classification, we investigate geometrical properties: curvature, Ricci tensor, sectional curvature, holonomy and parallel vector fields. The classification of algebraic Ricci solitons is also presented, as well as classification of pseudo-Kahler and ppwave metrics. We get the description of parallel symmetric tensors for each metric and showthat they are derived from parallel vector fields. Finally, we investigate the totally geodesic subalgebras by showing that for any subalgebra of the observed algebra there exists a metric that makes it totally geodesic.

math.DG↗

arXiv2021

Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

It is well known that $\mathbb{C}H^n$ has the structure of solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that all of these metrics are of constant negative scalar curvature and only one of them is Einstein (up to isometry and scaling). Finally, we present the relation between Ricci solitons on Heisenberg group and Einstein metric on $\mathbb{C}H^n$.

math.DG↗

arXiv2003

Para-quaternionic reduction

The pseudo-Riemannian manifold $M=(M^{4n},g), n \geq 2$ is para-quaternionic K\" ahler if $hol(M) \subset sp(n, \RR) \oplus sp(1, \RR).$ If $hol(M) \subset sp(n, \RR),$ than the manifold $M$ is called para-hyperK\" ahler. The other possible definitions of these manifolds use certain parallel para-quaternionic structures in $\End (TM),$ similarly to the quaternionic case. In order to relate these different definitions we study para-quaternionic algebras in details. We describe the reduction method for the para-quaternionic K\" ahler and para-hyperK\" ahler manifolds and give some examples. The decomposition of a curvature tensor of the para-quaternionic type is also described.

math.DG↗

arXiv2002

Examples of Self-dual, Einstein metrics of $(2,2)$-signature

In this paper we construct a family of examples of self-dual Einstain metrics of neutral signature, which are not Ricci flat, nor locally homogenous. Curvature of these manifolds is studied in details. These are obtained by the para-quaternionic reduction. We compare our examples with the orbifolds $\oo$ given by Galicki and Lawson, for which some new properties are also established. Particularly, the sign and the pinching of their sectional curvatures are studied.

math.DG↗