Killing fields generated by multiple solutions to the Fischer–Marsden equation

Abstract

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.

Killing fields generated by multiple solutions to the Fischer–Marsden equation II

“…Counterexamples to that have been found (see, for instance, Kobayashi [7] and Lafontaine [9] as well as our first paper [3]). We notice that, in almost all the known examples, the dimension of the solution space of (7) is at least 2.…”

“…One can prove that the space of Fischer-Marsden solutions has dimension less than or equal to (n + 1) with equality only if (M, g) has constant curvature (the dimension bound also appeared in [2, Corollary 2.4], but, it did not mention what happens when the dimension achieves the upper bound n + 1 there). In fact, we proved in our first paper [3], the following stronger statement. Therefore, if dim W = n, then the SO(dim W ) orbits are either S n−1 or fixed points.…”

“…By choosing the right orthonormal solutions x and y in Proposition 4 in our first paper [3], we might assume that Y x = βy = −y and Y y = −βx = x, i.e. β = −1.…”

A Note on the Classification of Compact Homogeneous Locally Conformal Kähler Manifolds

“…Although my proofs might be simpler and more convincing, they were hesitated to change their original proof. These results were also announced in (Cernea and Guan, 2015). Although Theorem 1 might be kind of trivial on the group level by the Annals paper (Hochchild and Serre, 1953), it was not very clear to us that it is also true for the coset space.…”

“…We list some good references as (Borel and Remmert, 1962;Chevalley, 1968;Cernea and Guan, 2015;Dorfmeister and Guan, 1991;Dorfmeister and Nakajima, 1988;Guan, 2002;1996;Gauduchon et al, 2015;Hasegawa and Kamishima, 2016;Hano and Kobayashi, 1960;Hochchild and Serre, 1953;Koszul, 1955;Mostow, 1961;Matsushima, 1957;Nakamura, 1975;Nomizu, 1954;Tits, 1971;Wang, 1954) as examples at the end. From our Theorem 1, we also obtained that the semisimple part S of the Lie group acts on M with cohomogeneity one action, i.e., with hypersurface orbits.…”