Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.02062

Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

Vicente Cortés·Shubham Dwivedi·Lorenz Jetter
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that on a connected, simply-connected nilpotent Lie group, the notions of left-invariant generalized Ricci soliton, semi-algebraic and algebraic generalized Ricci soliton coincide. Moreover, every such soliton has harmonic torsion. We classify all algebraic solitons with positive definite metric on the five-dimensional Heisenberg algebra, the four-dimensional oscillator algebra and the four-dimensional split-oscillator algebra up to automorphisms and rescalings. We find that there is only a finite number of algebraic solitons, all of which have harmonic torsion.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vicente Cortés, Shubham Dwivedi, Lorenz Jetter. 2026-11-01. Generalized Ricci Solitons on Heisenberg and Oscillator Algebras. https://arxiv.org/abs/2611.02062

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rotational symmetry of complete gradient Einstein-type manifolds with $α=0$ under vanishing conditions on several tensor fields

To provide a unified framework for various geometric solitons, Catino, Mastrolia, Monticelli, and Rigoli introduced gradient Einstein-type manifolds, which include gradient Yamabe solitons, gradient almost Yamabe solitons, gradient $k$-Yamabe solitons and conformal gradient solitons. In this paper, we show that any nontrivial complete four-dimensional gradient Einstein-type manifold with $α=0$, nonnegative scalar curvature, and vanishing Bach tensor is rotationally symmetric. The Bach-flat condition arises naturally in four-dimensional geometry and physics. We prove the main theorem by exploiting the conformal invariance of Bach-flatness and establishing a relation between two functionals. We also provide classification results under vanishing conditions on the Weyl tensor, the Cotton tensor, and the Cao-Chen-type tensor.

math.DG↗
arXiv2026

The min-max width of spheres associated to the distance function

What one obtains when the min-max methods for the distance function are applied on the space of pairs of points of a Riemannian two-sphere? This question is studied in details in the present article. We show that the associated min-max width do not always coincide with half of the length of a simple closed geodesic which is the union of two minimizing geodesics with the same endpoints. Therefore, it is a new geometric invariant. We study the structure of the set of minimizing geodesics joining a pair of points realizing the width, and relationships between this invariant and the diameter. The extrinsic case of an embedded Riemannian sphere is also considered.

math.DG↗
arXiv2026

On the $\mathcal{D}^+_J$ operator on higher-dimensional almost Kähler manifolds

In this paper, we introduce $\mathcal{D}^+_J$, a generalization of $\partial\bar{\partial}$ operator on higher dimensional almost Kähler manifolds. Using the $\mathcal{D}^+_J$ operator, we investigate the $\bar{\partial}$-problem in almost Kähler geometry and explore the generalized Monge-Ampère equation on almost Kähler manifolds. We establish uniqueness up to an additive constant. We also prove a local existence theorem for this equation on almost Kähler four-manifold. At last, we find an elliptical system for $\mathcal{D}^+_J$ operator. As an application, we reorganize the result of Tosatti-Weinkove-Yau in \cite{TWY}.

math.DG↗