Some problems on the geometry of calibrated manifolds

Yalçınkaya, Eyüp
In this thesis,w estudy three problems ont he geometry of calibrated manifolds,which are Riemannian manifolds equipped with a special closed differential form called a calibration. Firstly, we compute the homology of Grasmannian manifold of oriented 3-planes in R6, namely G+ 3 (R6), and its special submanifold called SLAG, the set of 3-planes in G+ 3 (R6) determined by the special Lagrangian calibration on CalabiYau 3-fold C3 ∼ = R6. We make an immediate application of these computations. Secondly,weinvestigatearelatedproblemontheembeddingoforientedclosedmanifoldsintoCn asspecialLagrangian-free(sLag-free). Finally,westudythegeography of symplectic 8-dimensional manifolds and obtain certain results on the existence of symplectic 8-manifolds with Spin(7)-structure.