# The conjugate of a subbundle of the complexified tangent bundle

Given a smooth manifold $M$ and consider the complexified tangent bundle $TM\otimes\mathbb C$. Let $HM$ be a subbundle of $TM\otimes\mathbb C$. What do we mean by $\overline {HM}$?

• There is a conjugation $\bar{(\cdot)} : T_xM \otimes \mathbb C\to T_xM \otimes \mathbb C$ sending $v\otimes z$ to $v\otimes \bar z$. I suppose $\bar{HM}$ is the image of $HM$ under this map. – user99914 Sep 4 '17 at 23:51
• Is the map $\bar{( \cdot)}$ has anything to do with the complex structure $J$ on $TM\otimes C$? – Ronald Sep 4 '17 at 23:57
• Which complex structure on $TM \otimes C$? – user99914 Sep 4 '17 at 23:58
• I am not sure actually. is there any complex structure on $TM\otimes C$ since it is complex vector bundle? – Ronald Sep 4 '17 at 23:59
• I see, so it is probably this one: $J(v\otimes z) = v\otimes (iz)$? – user99914 Sep 5 '17 at 0:03