Given an $m$-form $\omega \in \Omega^m(Y)$ in some $n$-dim manifold $Y$ and an inclusion map $\iota: X \rightarrow Y$, where $X$ is an $m$-dim submanifold embedded in $Y$, are there any properties inherited by the pullback of the $m$-form under the inclusion, $\iota^* \omega \in \Omega^m(X)$?
I think that since an embedding is a diffeomorphism onto its image, $\iota^* \omega$ should be a well-defined volume form on $X$. In particular, it seems reasonable that
$$ \int_X \iota^* \omega = \int_{X \subset Y} \omega, $$
but am not sure how to prove this.
Best Answer
Your last comment finally makes it clear that you're thinking about hermitian (or Kähler) geometry. It is a very beautiful and special situation in Kähler geometry that when $Y$ is a Kähler manifold with Kähler form $\omega$, it is the case that for any $m$-dimensional complex submanifold $X$ (the restriction of) $\omega^m/m!$ will in fact be the induced volume form of $X$. This follows from the Wirtinger inequality and is very special. There is no such fact in the case of real manifolds. See further discussion.