On the Prescribed Ricci Curvature of Noncompact Homogeneous Spaces with Two Isotropy Summands

Abstract

The current dissertation works within the setting of noncompact homogeneous spaces 𝐺/𝐻 in which 𝐺 is semi-simple. In particular, we frequently work with a decomposition of the Lie algebra 𝔤, 𝔤 = 𝔥 ⊕ 𝔭'' ⊕ 𝔭', where 𝔥 ⊕ 𝔭'' is the maximal compact in 𝔤 and 𝔭' is the negative one eigenspace from the Cartan decomposition. In such a setting we primarily set out to understand 𝐺 invariant metrics and Ricci curvature, and the relationship these are in with Lie theoretic conditions. There are three basic components to this work with the second holding most of our attention. The first component is an investigation into spaces, 𝐺/𝐻, in which we can always obtain some decomposition with (𝔭'', 𝔭') = 0 (what we call a Cartan orthogonal pair), building out results indicating that there are many examples of such spaces. The second component is an investigation into simply connected 𝐺/𝐻 with two isotropy irreducible summands. Here, we classify such spaces and solve the so-called Prescribed Ricci Curvature problem for all such 𝐺/𝐻. The third component is an investigation into a particularly nice setting of 𝐺/𝐻 with 𝐺 simple and having three irreducible summands in which [𝔭_𝑖, 𝔭_𝑖] ⊂ 𝔥 for each irreducible isotropy representation, 𝔭_𝑖. Here, we provide Lie theoretic conditions for obtaining diagonal 𝑟𝑖𝑐, begin an investigation into the signature of such spaces, and work through an example, 𝑆𝑂(𝑛, 2)/𝑆𝑂(𝑛). A final consequence of these three components is a description of the signature of all spaces 𝐺/𝐻 in which 𝐺 is simple and 𝐺/𝐻 has negative scalar curvature for all metrics.