Let $M$ be a complex manifold, $N$ is a smooth immersed submanifold of $M$. If $T\_p M$ is invariant under the multiplication by $i$ for any $p\in M$, then can we conclude that $N$ is a complex immersed submanifold of $M$?


Since $C^1$ property somehow means analytic property in complex setting, can we drop the assumption to that $N$ is merely a $C^1$ submanifold?



Smooth submanifold of a complex manifold with invariant tangent space under multiplication by $i$
Let $f : N \to M$ denote the immersion.
Since $f\_\*TN$ is invariant under $I$ where $I$ is the underlying almost complex structure of $M$, it induces an almost complex structure $I'$ on $N$.
Applying Newlander-Nirenberg theorem, we see that since
$I$ is integrable, $I'$ is also integrable.
Thus $I'$ is a complex structure on $N$, and
$f: (N,I') \to (M,I)$
is holomorphic.


