O ct 2 00 5 Correction and improvement added to “ Isospectral pairs of metrics on balls and spheres with different local geometries ”

The intertwining operator, κ, in the isospectrality proof of [Sz1, Sz2] 1 2 3 does not appear in the right form. The correct operator associates rather the Z-Fourier transforms of the functions corresponded to each other by this κ. In this note the isospectralities are established by means of this redefined operator. New isospectrality examples are also constructed. They live on sphere×balland sphere×sphere-type manifolds. They include isospectral metrics such that one of them is homogeneous while the other is locally inhomogeneous. Such examples were found on sphere×torusand sphere-type manifolds earlier. Operator κ is defined by κ : φ(|X|, Z)ΘpQΘ q Q → φ(|X|, Z)Θ ′p QΘ ′q Q. Although, the emphasis should have been placed on eigenfunctions with real eigenvalues of operator M = ∑l α=1 ∂αDα•, in this definition one focused on eigenfunctions with imaginary eigenvalues of operators DA• and DA′•. There have been overlooked that the κ is not complex linear, thus it does not intertwine them, as opposed to the rest parts of the Laplacians which are intertwined. The right operator corresponds the partial Z-Fourier transforms, ∫ z e φΘpQΘ q QdV and ∫ z e 〉φΘ QΘ ′q QdV , of the above functions to each other. All the right elements of that proof are used also here, however, these considerations penetrate more deeply into explicit spectrum computations. Annals of Mathematics, 154(2001), 437-475; and 161(2005), 343-395 1991 Mathematics Subject Classification. Primary 58G25, 53C20