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Description:
In this dissertation we study a conjecture attributed to Chern and Hopf, a generalization of Atiyah's L 2 -index theorem used by Gromov to prove said conjecture for Kähler manifolds, and an extension to Kähler orbifolds and all known examples of quaternionic Kähler manifolds. The study of a two-form η with values in a bundle G over a quaternionic Kähler manifold motivates us to show vanishing results for the cohomology of so 5 (C)-modules. We use these results to show that in certain degrees all G-valued d ∇ -closed forms are d ∇ -exact.