We have a complex manifold $M$ equiped with a hermitian metric, then for a complex submanifold $S \subset W$, the Wirtinger's theorem tells us that the volume form on $S$ is the restriction of a global form on $M$.
The textbook then made the remark that this is not true in the real case. Does anyone have an explicit example to explain that?
Explicit example: Let $M = \mathbb R^2$. Let $\alpha$ be any one form on $\mathbb R^2$. Write $$\alpha = Adx + Bdy,$$ where $A, B$ are smooth functions. Consider the vector fields $X(x, y) = (B, -A)$ and let $\gamma(t)$ be any intgral curve of $X$. Then $$\gamma^*\alpha = 0$$ and thus $\alpha$ do not restrict to the volume form on $\gamma$.