Generalized Hermann theorems and conformal holomorphic submersions
In this paper, the author considers submersions between Riemannian manifolds and proves: (i) If \(\nu\) is a holomorphic submersion of Kähler manifolds then \(\nu\) is Riemannian if and only if it is flat. (ii) If a holomorphic flat submersion \(\nu\) with totally geodesic fibers is defined on a Kähler manifold \(E\), then \(E\) is locally reducible. We observe that in [J. Differ. Geom. 11, 147-166 (1976; Zbl 0355.53037)] \textit{B. Watson} has proved that if \(\nu:E\to M\) is a Kähler submersion with totally geodesic fibres then \(\nu\) is a smooth fibre bundle with flat connection and \(E\) is a locally product manifold. Also the author studies conformal and holomorphic submersions and he obtains sufficient conditions for a holomorphic conformal submersion of a Kähler manifold to be the projection of a holomorphic fiber bundle with isomorphic fibers. These results generalize those obtained by \textit{R. Hermann} in [Proc. Am. Math. Soc. 11, 236-242 (1960; Zbl 0112.13701)] for Riemannian submersions. Finally, he gives a method for constructing these bundles with totally geodesic fibers.
scientific article; zbMATH DE number 1429654
scientific article