Holomorphic vector bundles on K\"ahler manifolds and totally geodesic foliations on Euclidean open domains

TL;DR

This paper explores the connection between holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean domains, providing geometric conditions and recovering known results for codimension-two cases.

Contribution

It establishes a new relation between sections of holomorphic vector bundles and geodesic foliations, using orthogonal Grassmannians to characterize when foliations originate from bundles.

Findings

Identifies geometric conditions linking holomorphic bundles and foliations.

Recovers known results for codimension-two foliations.

Highlights the role of orthogonal Grassmannians in this context.

Abstract

In this Note we establish a relation between sections in globally generated holomorphic vector bundles on K\"ahler 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. For codimension-two foliations, this description recovers of P. Baird and J. C. Wood. The universal objects that play a key role are the orthogonal Grassmannians.