Title: 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 Kähler 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.