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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.

Eweler
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    Do you mean an integral over $Y$ on the RHS? Also, note that volume forms by definition have to be nowhere vanishing, but if you pullback a form to some submanifold it could become 0 (even identically) hence not a volume form (e.g in $\Bbb{R}^3$ consider the 2-form $dx\wedge dy$, which if pulled back to a plane of constant $y$, becomes 0). – peek-a-boo Jul 25 '22 at 14:43
  • You misspeak when you say "the pullback of the volume form." The volume form on $Y$ is an $n$-form, not an $m$-form. But a good question is this: How do you know there is an $m$-form on $Y$ whose restriction to $X$ is the volume form of $X$? For any $m$-form $\omega$ on $Y$, the equation $$\int_X \iota^*\omega = \int_{\iota(X)}\omega$$ is a tautology. – Ted Shifrin Jul 25 '22 at 16:52
  • Thanks for the responses, I've edited my question slightly to make it less incorrect. I was thinking in the context of the Koidara embedding of an $n$-dim hypersurface $X$ in $\mathbb{CP}^{n+1}$ which is realized by the zero locus of a degree $n+2$ homogeneous polynomial. Using the natural Fubini-Study form $\omega_{FS}$ in $\mathbb{CP}^{n+1}$, one may construct an $(m,m)$-form $\omega_{FS}^m$ - which according to @TedShifrin 's comment satisfies

    $$ \int_X \iota^* \omega_{FS}^m = \int_{\iota(X)} \omega_{FS}^m $$.

     – Eweler Jul 26 '22 at 10:30

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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.

Ted Shifrin
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