Journal of the
Korean Mathematical Society JKMS

In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let $\varphi: (M^n, g) \to (N^m, h)$ 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 $\varphi$. 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, $\varphi^{-1}(P)$, 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.

Keywords: harmonic morphism, horizontally homothetic, stable minimal submanifold, totally geodesic

MSC numbers: Primary 53C43, 58E20