On the spectral geometry for the Jacobi operators of harmonic maps into Kenmotsu manifolds (Q2757179)

This article establishes rigidity results for the spectrum of the Jacobi operator of harmonic maps from a Riemannian manifold into a Kenmotsu manifold.NEWLINENEWLINENEWLINEHarmonic maps between Riemannian manifolds are critical points of the energy functional and the second variation operator is called the Jacobi operator \(J\). It is the sum of the rough Laplacian of the pull-back bundle and a curvature term.NEWLINENEWLINENEWLINEAs for the heat operator, one can consider the asymptotic expansion of the semi-group associated to \(J\), and, following P. Gilkey, \textit{H. Urakawa} [Ill. J. Math. 33, No. 2, 250-267 (1989; Zbl 0644.53048)] computed the first few coefficients. NEWLINENEWLINENEWLINEWhen the target is a Kenmotsu manifold, i.e. an almost contact metric manifold with an adapted connection, and its \(\phi\)-sectional curvature is constant, then the special form of the curvature tensor yields simple expressions for the first three coefficients. In particular, the energy of the map appears explicitly in \(a_1\).NEWLINENEWLINENEWLINEIsometric immersions into a Kenmotsu manifold are classified in three types: invariant, tangential anti-invariant and normal anti-invariant, according to the behaviour of the image of tangent planes and the position of the structure vector field.NEWLINENEWLINENEWLINEThen, it is shown that, within either of these classes, a map with the same spectrum as a totally geodesic map, must be totally geodesic as well. Moreover, if two isometric minimal immersions share the same spectrum, they must belong to the same class.