Geodesic
On skew 2-projectable almost complex structures on $TM$
Archivum mathematicum, Tome 34 (1998) no. 2, pp. 285-293 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\nabla _{\Gamma _{\alpha }} (J\alpha ) =0$.
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 $\nabla _{\Gamma _{\alpha }} (J\alpha ) =0$.
Classification : 53C05, 53C15, 58A20
Keywords: tangent bundle; skew 2-projectable; $(1, 1)$-vector fields; almost complex structure; connection 
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     author = {Dekr\'et, Anton},
     title = {On skew 2-projectable almost complex structures on $TM$},
     journal = {Archivum mathematicum},
     pages = {285--293},
     year = {1998},
     volume = {34},
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     mrnumber = {1645324},
     zbl = {0910.53020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1998_34_2_a5/}
}
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Dekrét, Anton. On skew 2-projectable almost complex structures on $TM$. Archivum mathematicum, Tome 34 (1998) no. 2, pp. 285-293. http://geodesic.mathdoc.fr/item/ARM_1998_34_2_a5/

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