Characterization of totally geodesic foliations with integrable and parallelizable normal bundle

Abstract

Abstract In this work, we study foliations of arbitrary codimension $\mathfrak{F}$ with integrable normal bundles on complete Riemannian manifolds. We obtain a necessary and sufficient condition for $\mathfrak{F}$ to be totally geodesic. For this, we introduce a special number $\mathfrak{G}_{\mathfrak{F}}^{\alpha}$ that measures when the foliation ceases to be totally geodesic. Furthermore, applying some maximum principle we deduce geometric properties for $\mathfrak{F}$ . We conclude with a geometrical version of Novikov’s theorem (Trans. Moscow Math. Soc. (1965), 268–304), for Riemannian compact manifolds of arbitrary dimension.