The prescribed Ricci curvature problem for naturally reductive metrics on non-compact simple Lie groups

Abstract

We investigate the prescribed Ricci curvature problem in the class of left-invariant naturally reductive Riemannian metrics on a non-compact simple Lie group. We obtain a number of conditions for the solvability of the underlying equations and discuss several examples.

On the variational properties of the prescribed Ricci curvature functional

“…By continuity, there exists a neighborhood in M T of the green line segment T 1 = T 2 < 1 4 in Figure 5 where S |M T has a saddle. We point out that (7.20) is the Jensen Einstein metric when t = 3 4 , and a local maximum. It is marked with the blue dot in Figure 5.…”

“…In the case where the quotient M/G is onedimensional, the problem was addressed by Hamilton [21], Cao-DeTurck [13], Pulemotov [29,30] and Buttsworth-Krishnan [10]. The case where M is a homogeneous space G/H has been studied extensively; see the survey [11] and the more recent references [12,25,26,4,3]. In some situations, the equation can be solved explicitly, as shown, e.g., in [31,9].…”

“…, a Riemannian metric with positivedefinite Ricci curvature for t ∈3 7 , 1 , and a Riemannian metric with indefinite Ricci curvature for t ∈…”