Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

In this Note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a non-degenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in a holomorphic vector bundle. This description recovers characterisations of Baird and Wood for Euclidean 3-space. The universal objects that play a key role are the orthogonal Grassmannians.