Quaternionic analysis on Riemann surfaces and differential geometry
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.
scientific article; zbMATH DE number 1184338