Direct proof that a parallel almost complex structure is integrable
$begingroup$
Let $(M,g,J)$ be an almost Hermitian manifold nad $nabla$ be the Levi-Civita connection on $M$. If $nabla J=0$, it is straightforward to show that the Nijenhuis tensor of $J$ must vanish which implies that $J$ is integrable. But the condition $nabla J=0$ is stronger than $N_J=0$, so my question is: Is there a direct proof of this statement without invoking the Newlander-Nierenberg theorem?
differential-geometry complex-geometry kahler-manifolds almost-complex
$endgroup$
add a comment |
$begingroup$
Let $(M,g,J)$ be an almost Hermitian manifold nad $nabla$ be the Levi-Civita connection on $M$. If $nabla J=0$, it is straightforward to show that the Nijenhuis tensor of $J$ must vanish which implies that $J$ is integrable. But the condition $nabla J=0$ is stronger than $N_J=0$, so my question is: Is there a direct proof of this statement without invoking the Newlander-Nierenberg theorem?
differential-geometry complex-geometry kahler-manifolds almost-complex
$endgroup$
add a comment |
$begingroup$
Let $(M,g,J)$ be an almost Hermitian manifold nad $nabla$ be the Levi-Civita connection on $M$. If $nabla J=0$, it is straightforward to show that the Nijenhuis tensor of $J$ must vanish which implies that $J$ is integrable. But the condition $nabla J=0$ is stronger than $N_J=0$, so my question is: Is there a direct proof of this statement without invoking the Newlander-Nierenberg theorem?
differential-geometry complex-geometry kahler-manifolds almost-complex
$endgroup$
Let $(M,g,J)$ be an almost Hermitian manifold nad $nabla$ be the Levi-Civita connection on $M$. If $nabla J=0$, it is straightforward to show that the Nijenhuis tensor of $J$ must vanish which implies that $J$ is integrable. But the condition $nabla J=0$ is stronger than $N_J=0$, so my question is: Is there a direct proof of this statement without invoking the Newlander-Nierenberg theorem?
differential-geometry complex-geometry kahler-manifolds almost-complex
differential-geometry complex-geometry kahler-manifolds almost-complex
asked Jan 18 at 14:23
deepfloedeepfloe
714
714
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