Chapter 2 Spaces of metrics and curvature functionals

Abstract

This chapter discusses the spaces of metrics and curvature functionals. Two C ∞ metrics on the same manifold may well have very different curvature properties; the case of a bump in the plane can be considered. The component functions g ij of two metrics may be C 0 or C 1 close but not C 2 close, so their curvatures would not be C 0 close. Similarly, the curvature of the sum of two metrics is not related to the curvature of the individual summands. The chapter presents a result of H. Wu on Hermitian metrics. The functional A(g) , B(g) , and D(g) are discussed in the chapter. Metrics of constant curvature and Kahler metrics of constant holomorphic curvature are critical points of D(g). Also, a canonical invariant metric on a compact semisimple Lie group is a critical point of D(g). The chapter discusses the subspaces of metrics associated to a symplectic or contact structure and to other functionals.