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On skew 2-projectable almost complex
structures on TM

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*
Anton Dekret
*

** Address.**
Department of Mathematics, TU Zvolen,
Masarykova 24, 960 53, Zvolen, SLOVAKIA

** E-mail:**
dekret@vsld.tuzvo.sk

**Abstract.**
We deal with a $(1, 1)$-tensor field $\alpha$ on the
tangent bundle $TM$ preserving vertical vectors and such that
$J\alpha =-\alpha J$ is a $(1, 1)$-tensor field on $M$, where $J$
is the canonical almost tangent structure on $TM$. A connection
$\Gamma _{\alpha}$ on $TM$ is constructed by $\alpha$. It is
shown that if $\alpha$ is a $VB$-almost complex structure on $TM$
without torsion then $\Gamma _{\alpha}$ is a unique linear
symmetric connection such that $\alpha (\Gamma _{\alpha})=\Gamma
_{\alpha}$ and $\nabla _{\Gamma _{\alpha}} (J\alpha) =0$.

**AMSclassification.**
53C05, 58A20

**Keywords.**
Tangent bundle, skew 2-projectable, $(1, 1)$-vector
fields, almost complex structure, connection