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On Jacobi Fields Along Eigenmappings of the Tension Field for Mappings into a Symmetric Riemannian Manifold

Volume 18, Number 1 (2015), 98 - 121

On Jacobi Fields Along Eigenmappings of the Tension Field for Mappings into a Symmetric Riemannian Manifold

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Abstract

We prove that the mean value ( for some measure $\mu =\chi dx$ with $\chi \geq 0,dx=$ Riemannian measure ) of the squared norm of the gradient of the unitary direction of a Jacobi field along an eigenmapping $v$ ( associated to an eigenvalue $\lambda \geq 0$ ) of the tension field, for mappings from a compact Riemannian manifold $(M,g)$ into a symmetric Riemannian manifold $(N,h)$ of positive sectional curvature, is smaller than $c\lambda $, where $c>0$ depends only on the diameter and upper and lower curvature bounds of $(N,h)$. For negative $\lambda $, we prove that there is no nonvanishing Jacobi field along the eigenmappings, under the same assumptions on $(M,g)$ and $(N,h)$.