SPECTRAL GEOMETRY OF HARMONIC MAPS INTO WARPED PRODUCT MANIFOLDS WITH A CIRCLE JEONGWOOK CHANG AND GABJIN YUN

Abstract. Let (Mn, g) be a closed Riemannian manifold and N = S1 ×f Sm−1c be a warped product space with the warped product metric h = dt2 + f2(t)ds2. Let φ, ψ:M → N be two base projectively harmonic maps. We prove that if the Jacobi operators Jφ and Jψ of φ and ψ are isospectral, then the energy of φ and ψ are equal up to constant. Besides we show some properties of harmonic maps and its relation with the spectral geometry of warped product Riemannian manifolds with a circle. In this paper, we deal with the inverse spectral problem of the Jacobi operator of a harmonic map from a compact manifold into a warped product manifold. The relationship between the geometry of a smooth manifold and the spectrum of the Laplacian has been studied by many authors ([1], [5], [6], etc.). In [5], P