African Diaspora Journal of Mathematics

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

Moussa Kourouma

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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 $(N,h)$.

Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 18, Number 1 (2015), 98-121.

Dates
First available in Project Euclid: 2 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.adjm/1446472409

Mathematical Reviews number (MathSciNet)
MR3399810

Zentralblatt MATH identifier
1328.58006

Subjects
Primary: 58C 58E 58J 49R50 35J 35D

Keywords
Riemannian manifold tension field Jacobi field eigenvalue eigenmapping convexity symmetry

Citation

Kourouma, Moussa. On Jacobi Fields Along Eigenmappings of the Tension Field for Mappings into a Symmetric Riemannian Manifold. Afr. Diaspora J. Math. (N.S.) 18 (2015), no. 1, 98--121. https://projecteuclid.org/euclid.adjm/1446472409


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