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tangent bundle; skew 2-projectable; $(1, 1)$-vector fields; almost complex structure; connection
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 $\nabla _{\Gamma _{\alpha }} (J\alpha ) =0$.
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