On skew 2-projectable almost complex structures on TM

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