Timeline for $SU(n)$-structures on a manifold

5 events

when toggle format	what		by	license	comment
Mar 19, 2013 at 19:08	comment	added	Robert Bryant		(cont) Oh, and, with those hypotheses, you don't have to assume that $M$ has an almost complex structure, the form $\Omega$ satisfying the conditions (i) and (ii) define an almost complex structure on $M$, namely, the one for which $\Omega$ has type $(n,0)$.
Mar 19, 2013 at 19:04	comment	added	Robert Bryant		Even with the right definition of $\mathrm{SU}(n)$-structure (which, as Ben points out, you don't have), you'll need more hypotheses than you have given. For example, your hypotheses allow $\omega=0$, which you surely don't want. You need (i) $\Omega$ is decomposable, (ii) $\Omega\wedge\bar\Omega\not=0$, and (iii) when you write $\Omega = \theta^1\wedge\cdots\wedge\theta^n$, then $\omega = (i/2)\ g_{k\bar l}\ \theta^k\wedge\overline{\theta^l}$ where $g = (g_{k\bar l})= (\ \overline{g_{l\bar k}}\ )$ is Hermitian positive definite and has determinant $1$.
Mar 19, 2013 at 18:03	answer	added	Ben McKay		timeline score: 4
Mar 19, 2013 at 17:18	history	edited	hapchiu	CC BY-SA 3.0	added 15 characters in body
Mar 19, 2013 at 16:36	history	asked	hapchiu	CC BY-SA 3.0