Manifolds of Riemannian Metrics with Prescribed Scalar Curvature by Arthur E. Fischer and Jerrold

THEOREM 2. Assume J * V 0 . Writing UtQ=(je a o\0 )U&9 J(\ is the disjoint union of closed submanifolds. REMARK. If d i m M = 2 , e^J=^" 8 , and if d i m M = 3 , the hypothesis that 1F*J£0 can be dropped. The proof of Theorem 1 also allows us to conclude that a solution h of the linearized equations DR(g0) • h=0 is tangent to a curve of exact solutions of R(g)=p through a given solution g0, provided p is not a constant ^ 0 . In the terminology of [4] we say the equation R(g)=p is linearization-stable at g0. From Theorem 3 below the equation R(g)=0 is still linearization-stable about a solution g0 provided Ric(g0) is not identically zero. For the singular case p = 0 , Theorem 2 incorporates an isolation theorem inspired by the work of Brill and Deser [2], namely, that the flat metrics are isolated solutions of R(g)=0. As a corollary one has: If g(t) is a