In Memory of Professor S. KobayashiIn the process of finding Einstein metrics in dimension n ≥ 3, we can search critical metrics for the scalar curvature functional in the space of the fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer-Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer-Marsden conjecture said that if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by O. Kobayashi and J. Lafontaine. However, almost all the counter-examples are homogeneous. Multiple solutions to this system yield Killing vector fields. We show that the dimension of the solution space W can be at most n + 1, with equality implying that (M, g) is a sphere with constant sectional curvatures. Moreover, we show that the identity component of the isometry group has a factor SO(W ). We also show that geometries admitting FischerMarsden solutions are closed under products with Einstein manifolds after a rescaling. Therefore, we obtain a lot of non-homogeneous counter-examples to the Fischer-Marsden conjecture. We then prove that all the homogeneous manifold M with a solution are in this case. Furthermore, we also proved that a related Besse conjecture is true for the compact homogeneous manifolds.

In the process of finding Einstein metrics in dimension [Formula: see text], we can search metrics critical for the scalar curvature among fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that, if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by Kobayashi and Lafontaine, and by our first paper. Multiple solutions to this system yield Killing vector fields. We showed in our first paper that the dimension of the solution space [Formula: see text] can be at most [Formula: see text], with equality implying that [Formula: see text] is a sphere with constant sectional curvatures. Moreover, we also showed there that the identity component of the isometry group has a factor [Formula: see text]. In this second paper, we apply our results in the first paper to show that either [Formula: see text] is a standard sphere or the dimension of the space of Fischer–Marsden solutions can be at most [Formula: see text].