| Popis: |
The starting point of the thesis is the definition of an almost quaternionic manifold as a real 4n---manifold admitting a GL(n,H) Sp(1)-structure. It is showh that there exist two obstructions C0,C1, to the integrability of such a structure, and the condition C0 = 0 is used to define a cuaternionic manifold. It is proved that a_ quaternionic manifold M has a natually associated complex (2n+1)-manifold Z fibring over M with fibre CP1, and the relationship between the complex structure of Z and the quaternionic structure of M- is studied. The above theory is then applied to the case in which M is a Riemannian manifold with holonomy contained in Sp(n)Sp(l), using properties of Z to gain information about M. Special emphasis is given to the situation when M has positive scalar curvature and it is shown that M is then isometric to quaternionic projective space HPn provided a certain mod 2 cohomolocly class vanishes. When M is 8-dimensional, estimates for its Betti numbers and isometry group are obtained. |
