Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds

PROPOSITION 3.4.Let M be a compact connected C°° manifold of dimension not less than two.If a Riemannian metric g belongs to the set S^i.e., if all the eigenvalues of the Laplacian A g have multiplicity one, then the group of all isometries of (M, g) is discrete.Combining this with Theorem 3.1, we have: COROLLARY 3.5.Let M be a compact connected G°° manifold of dimension not less than two.Then the set of all elements g in ^^ with discrete isometry group contains a residual subset of ^£.That is, for most Riemannian metrics of a compact connected C°°m anifold of dimension not less than two, the isometry groups are trivial.This corollary was obtained by Ebin (cf.[E^ Proposition 8.3]) in a different manner.