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 conculde that $N$ is a complex immersed submanifold of $M$?

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

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?