Abstract
In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let φ : (Mn, g) → (Nm, h) be a horizontally homothetic harmonic morphism from a Riemannian manifold into a Riemannian manifold of non-positive sectional curvature and let T be the tensor measuring minimality or totally geodesics of fibers of φ. We prove that if T is parallel and the horizontal distribution is integrable, then for any totally geodesic submanifold P in N , the inverse set, φ−1(P ), is volume-stable in M . In case that P is a totally geodesic hypersurface, the condition on the curvature can be weakened to Ricci curvature.
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