On Geometry of Manifolds with Some Tensor Structures and Metrics of Norden Type

Abstract

The object of study in the present dissertation are some topics in differential geometry of smooth manifolds with additional tensor structures and metrics of Norden type. There are considered four cases depending on the dimension of the manifold: 2n, 2n + 1, 4n and 4n + 3. The studied tensor structures, which are counterparts in the different related dimensions, are the almost complex/contact/hypercomplex structure and the almost contact 3-structure. The considered metric on the 2n-dimensional case is the Norden metric, and the metrics in the other three cases are generated by it. The purpose of the dissertation is to carry out the following: 1. Further investigations of almost complex manifolds with Norden metric including studying of natural connections with conditions for their torsion and invariant tensors under the twin interchange of Norden metrics. 2. Further investigations of almost contact manifolds with B-metric including studying of natural connections with conditions for their torsion and associated Schouten-van Kampen connections as well as a classification of affine connections. 3. Introducing and studying of Sasaki-like almost contact complex Riemannian manifolds. 4. Further investigations of almost hypercomplex manifolds with Hermitian-Norden metrics including studying of integrable structures of the considered type on 4-dimensional Lie algebra and tangent bundles with the complete lift of the base metric; introducing of associated Nijenhuis tensors in relation with natural connections having totally skew-symmetric torsion as well as quaternionic K\"ahler manifolds with Hermitian-Norden metrics. 5. Introducing and studying of manifolds with almost contact 3-structures and metrics of Hermitian-Norden type and, in particular, associated Nijenhuis tensors and their relationship with natural connections having totally skew-symmetric torsion.