# HORIZONTALLY HOMOTHETIC HARMONIC MORPHISMS AND STABILITY OF TOTALLY GEODESIC SUBMANIFOLDS

• Yun, Gab-Jin (DEPARTMENT OF MATHEMATICS MYONG JI UNIVERSITY) ;
• Choi, Gun-Don (GARC AND DEPARTMENT OF MATHEMATICS SEOUL NATIONAL UNIVERSITY)
• Published : 2008.03.31

#### Abstract

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

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#### Cited by

1. Stability and integrability of horizontally conformal maps and harmonic morphisms 2013, https://doi.org/10.1002/mana.201200183