Quaternionic analysis on Riemann surfaces and differential geometry
type: Item
instance of: scholarly article
title: Quaternionic analysis on Riemann surfaces and differential geometry
zbMATH Open document ID: 0910.53042
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/231230
full work available at URL: Pinkall.MAN.html
review text: The authors develop a new quaternionic approach for the study of conformal immersions \(f\) of a Riemannian surface \(M\) into \(\mathbb R^4\) and \(\mathbb R^3\). The starting point is the interpretation of the conformality of a map \(f : M \to \mathbb R^4 = \mathbb H \) as the condition \[ df_x \circ J = N_x \cdot df_x, \qquad x \in M, \] where \(J\) is the complex structure of \(M\), \(N : M \to S^2 = \{ q \in \mathbb H , q^2 =-1 \}\) is some map and dot means quaternion multiplication. The authors define the notion of holomorphic quaternion line bundle over \(M\) as the bundle \(L = E \oplus E\), where \(E\) is a holomorphic (complex) line bundle, together with a holomorphic structure. The holomorphic structure is defined as generalized Cauchy-Riemann operator \( D : \Gamma(L) \to \Gamma (K^* \otimes L)\), where \(K^*\) is the anticanonical bundle of \(M\), which is characterized by some properties. An example of such holomorphic structure is \(\overline \partial \oplus \overline \partial \). One of the main results of the theory is that quotients of holomorphic (i.e., belonging to \(\text{ker }D\)) sections of such holomorphic quaternionic line bundles are meromorphic conformal maps of \(M\) into \(\mathbb H\) and every conformal map can be obtained as such a quotient in a unique way.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 1184338
zbMATH Keywords: surfaces in \(\mathbb R^4\)
zbMATH Keywords: conformal immersion
zbMATH Keywords: conformal maps
zbMATH Keywords: quaternionic line bundles
zbMATH Keywords: holomorphic line bundles
zbMATH Keywords: Cauchy-Riemann equations
zbMATH Keywords: Willmore functional
zbMATH Keywords: quaternionic analysis
author: Franz Pedit
author: Ulrich Pinkall
reviewed by: Dmitri Alekseevsky
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184338