Riemannian submersions, minimal immersions and cohomology class

Abstract

Abstract: We prove a simple optimal relationship between Riemannian submersions andminimal immersions; namely, if a Riemannian manifold admits a non-trivial Riemannian submer-sion with totally geodesic fibers, then it cannot be isometrically immersed in any Riemannianmanifold of non-positive sectional curvature as a minimal manifold. Some related results are alsopresented. In the last section, we introduce a cohomology class for Riemannian submersions andprovide an application.Key words: Riemannian submersion; minimal immersion; cohomology class; totallygeodesic fibers.1. Introduction. Let Mand Bbe Rieman-nian manifolds with n= dimM >dimB= b>0.A Riemannian submersion π: M→ Bis a mappingof Monto Bsatisfying the following two axioms:(S1) πhas maximal rank;(S2) the differential π ∗ preserves lengths of horizon-tal vectors.The mappings between Riemannian mani-folds satisfying these two axioms were studied byT. Nagano in [10] in terms of fibred Riemannianmanifolds. In particular, he derived the fundamen-tal equations analogous to Weingarten’s formulas forRiemannian submanifolds. B. O’Neill further stud-ied such mappings in [11] and called them Rieman-nian submersions.For each p∈ B, π