Note on isospectral Riemannian manifolds
The author states two theorems in the paper. Theorem 1: Let \((M,g)\) and \((\bar M,\bar g)\) be two compact, connected, locally symmetric and conformally flat Riemannian manifolds. If they are isospectral, then they are isometric. Theorem 2: Let \((M,J,g)\) and \((\bar M,\bar J,\bar g)\) be two compact connected and locally symmetric Bochner Kähler manifolds. If they are isospectral, then they are holomorphically isometric. It is possible that these theorems are not precisely correct as stated. Vigneras has constructed examples of Riemann surfaces of constant sectional curvature \(-1\) which are isospectral but not isometric and in dimensions greater than 2 has constructed examples of manifolds of constant sectional curvature \(-1\) which are isospectral but not diffeomorphic. Ikeda has constructed examples of manifolds with constant sectional curvature \(+1\) which are isospectral but not isometric. Milnor has constructed examples of flat tori which are isospectral but not isometric. The examples all yield compact, connected, locally symmetric, and conformally flat manifolds which are isospectral but not isometric. The author at a crucial point in the paper cites a theorem of Ficken to assert that if \(M\) is a reducible locally symmetric and conformally flat manifold, then \(M\) is either the sphere \(S^ m(r)\) of radius \(r\) or is the product \(S^{m-1}(r)\) with the circle. The result (Theorem 3.3) Ficken actually proves is somewhat different and allows for spaces of arbitrary sectional curvature and for products of spaces of constant sectional curvature for which the sum of the curvatures vanishes.
scientific article; zbMATH DE number 436793