Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1511.08532

Algebraic Regularity over Quaternions and Regular Four-Manifolds

Keqin Liu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Based on a new generalization of Cauchy-Riemann system presented in this paper, we introduce a class of quaternion-valued functions of a quaternionic variable, which are called algebraic regular functions. The set of algebraic regular functions is not only a real associative algebra, but also respect the composition of functions. Using algebraic regular functions as transition maps, we introduce a class of four-manifolds called the regular four-manifolds.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Keqin Liu. 2015-11-27. Algebraic Regularity over Quaternions and Regular Four-Manifolds. https://arxiv.org/abs/1511.08532

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

KEEP EXPLORING

Related papers

arXiv2026

Uniform RC-positivity of tangent bundles

In this paper, we prove that every rationally connected projective manifold admits a smooth uniformly RC-positive Hermitian metric on its tangent bundle, answering Yang's question and giving a characterization of rational connectedness by uniform RC-positivity. We find an example to show that RC-positivity alone does not characterize rational connectedness. We also prove that uniform RC-positivity of the tangent bundle is preserved under blow-ups along connected smooth centers on compact complex manifolds. In the non-Kähler setting, we construct such metrics on all Hopf and Kato surfaces and obtain classification results for compact complex surfaces.

math.CV↗
arXiv2026

Growth, Distortion, and Schwarzian Norm Estimates for Exponentially Convex Functions

In this paper, we investigate the growth, distortion, pre-Schwarzian and Schwarzian norms of functions in the exponentially convex class \(\mathcal C_{e^λ}\), \(0<λ\leπ/2\), defined by \(1+zf''(z)/f'(z)\prec e^{λz}\). By representing the associated Schwarz function explicitly, we derive parameter-dependent estimates for \(f\), \(f'\), and the pre-Schwarzian derivative. We further obtain Schwarzian norm estimates under both the general normalization and the additional condition \(f''(0)=0\). The corresponding extremal problems are analyzed through suitable Schwarz functions, and the dependence of the resulting bounds on the exponential parameter is made explicit.

math.CV↗