= Smooth submanifolds with invariant complex tangent spaces are complex
{title2=$J(TY)=TY$}
Let $Y$ be a <embedded submanifold> of a <complex manifold>. If its <tangent spaces> are invariant under the <almost complex structure induced by a complex atlas>, choose ambient <holomorphic coordinates> making its <tangent space> at one point $\mathbb C^k\times0$. The first $k$ coordinates restrict to a local real <diffeomorphism> by the <inverse function theorem>, so $Y$ is locally the graph of a smooth function $F$. Invariance of the tangent graph gives $dF(iv)=i\,dF(v)$, the <Cauchy-Riemann equations>. Thus $F$ is <holomorphic> and these graph coordinates give the induced <complex submanifold> structure. This proves the result directly, without needing the general integrability theorem as an unproved step.
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