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.
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- OU - Dissertations [9338]