HORIZONTALLY HOMOTHETIC HARMONIC MORPHISMS AND STABILITY OF TOTALLY GEODESIC SUBMANIFOLDS

Title & Authors
HORIZONTALLY HOMOTHETIC HARMONIC MORPHISMS AND STABILITY OF TOTALLY GEODESIC SUBMANIFOLDS
Yun, Gab-Jin; Choi, Gun-Don;

Abstract
In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let $\small{\varphi:(M^n,g)\rightarrow(N^m,h)}$ 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 $\small{\varphi}$. 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, $\small{\varphi^{-1}}$(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;
Language
English
Cited by
1.
Stability and integrability of horizontally conformal maps and harmonic morphisms, Mathematische Nachrichten, 2013, n/a
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