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On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

100%
Bogdanovich S., Ermolitski A.
Open Mathematics
|
2004
|
tom 2
|
nr 5
615-623
EN
Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection $$\tilde \nabla $$ on an almost Hermitian manifold M an almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h 1 and h 2 are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold.
2
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On Bochner flat para-Kählerian manifolds

86%
Łuczyszyn D.
Open Mathematics
|
2005
|
tom 3
|
nr 2
331-341
EN
Let B be the Bochner curvature tensor of a para-Kählerian manifold. It is proved that if the manifold is Bochner parallel (∇ B = 0), then it is Bochner flat (B = 0) or locally symmetric (∇ R = 0). Moreover, we define the notion of tha paraholomorphic pseudosymmetry of a para-Kählerian manifold. We find necessary and sufficient conditions for a Bochner flat para-Kählerian manifold to be paraholomorphically pseudosymmetric. Especially, in the case when the Ricci operator is diagonalizable, a Bochner flat para-Kählerian manifold is paraholomorphically pseudosymmetric if and only if the Ricci operator has at most two eigenvalues. A class of examples of manifolds of this kind is presented.
3
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On the bochner conformal curvature of Kähler-Norden manifolds

86%
Olszak K.
Open Mathematics
|
2005
|
tom 3
|
nr 2
309-317
EN
Using the one-to-one correspondence between Kähler-Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics were established in my previous paper [19]. In the presented paper, we prove that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related. Especially, we find necessary and sufficient conditions for the holomorphic Weyl conformal curvature tensor of a Kähler-Norden manifold to be holomorphically recurrent.
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