UC Berkeley

In this thesis, we study the topology and geometry of homogeneous spaces of the form $G/T^k$, where $G$ is a compact semisimple Lie group, $T^k$ is an embedded torus in $G$. We say that a principal $Spin(n)$-bundle $P\rightarrow M$ admits a string structure if the structure group lifts to $String(n)$. In particular, a spin manifold is string if the principal $Spin(n)$-bundle associated to the tangent bundle has a string structure. It's known that $G/T^k$ are string manifolds and we prove the uniqueness of string structures on $G/T^k$ when $G$ is simply connected . There is a canonical metric on $G/T^k$, which has positive Ricci curvature. We deform this metric to a 1-parameter family of invariant metrics on $G/T^k$. We prove that the first Pontrjagin forms associated to the Levi-Civita connection of these metrics do not vanish. This verifies a conjecture by Redden-Stolz on the TMF-Witten genus of Ricci positive string manifolds.