I am trying to understand differential forms on manifolds with boundaries, and I am a bit confused with the boundary conditions.
For the following, let $(M,g_M)$ be a smooth Riemannian manifold with boundary $\partial M$, define $\iota:\partial M\hookrightarrow M$ the inclusion of the boundary, and $\star$ is the Hodge-star with respect to $g_M$.
In [1] the authors define the space of $p$-forms satisfying Dirichlet $(\Omega^p_{\text{D}}(M))$ and Neumann $(\Omega^p_{\text{N}}(M))$ boundary conditions. Their definition, towards the end of page 3, is equivalent to \begin{align} \Omega^p_{\text{D}}(M) &:= \Big\lbrace\omega\in\Omega^p(M)\ \Big\vert\ \iota^*\omega = 0\Big\rbrace \\ \Omega^p_{\text{N}}(M) &:= \Big\lbrace\omega\in\Omega^p(M)\ \Big\vert\ \iota^*\star\omega = 0\Big\rbrace .\end{align}
Verbosely, this defines Dirichlet $p$-forms, as $p$-forms that vanish upon eating vectors that are tangent to $\partial M$, and Neumann $p$-forms as $p$-forms that vanish upon eating vectors normal to $\partial M$.
Had I not seen that definition I would have defined Dirichlet $p$-forms in the same way, but the Neumann ones I would define by demanding that their derivative along the normal directions to $\partial M$ vanishes, in the same fashion as one does for functions. This translates to demanding that $$\iota^*\star\mathrm{d}\omega = 0.$$
My question is, is my definition of Neumann $p$-forms the same as [1]'s? I would expect not, as having vanishing $\star\mathrm{d}\omega$ is clearly less restrictive than having vanishing $\star\omega$. But then, if not, why is their definition justified as Neumann boundary conditions? I.e. how does it generalize the usual Neumann boundary conditions for functions in a natural way, and why is my naive definition wrong?
Moreover, a space that is important is the space $$\Omega^p_{\text{important}}(M) := \Big\lbrace\omega\in\Omega^p(M)\ \Big\vert\ \iota^*\mathrm{d}\omega = 0\Big\rbrace.$$ This space arises naturally in gauge theories as the space of gauge transformations that vanish on the boundary. How is this space related to $\Omega^p_{\text{D}}(M)$ or $\Omega^p_{\text{N}}(M)$? (apart from the obvious $\Omega^p_{\text{D}}(M)\subset\Omega^p_{\text{important}}(M)$)
References: [1] Sylvain Cappell, Dennis DeTurck, Herman Gluck, Edward Y. Miller, Cohomology of Harmonic Forms on Riemannian Manifolds With Boundary, https://arxiv.org/abs/math/0508372
This isn’t entirely an answer, rather a bit of intuition. The gist of it is that “Dirichlet” and “Neumann” collectively refer to boundary conditions which give rise to a well-posed elliptic system. In the case of the Hodge Decomposition Theorem, the relevant elliptic system is $$ \Omega^0 \overset{d}{\longrightarrow} \Omega^1 \overset{d}{\longrightarrow} \Omega^2 \overset{d}{\longrightarrow} \dotsm $$ The boundary condition arises by doing integration by parts: $$ \langle d\alpha, \beta \rangle = \int_M d\alpha \wedge \star\beta = \pm \int_M \alpha \wedge d\star\beta + \int_{\partial M} \alpha \wedge \star\beta . $$ ($\pm$ because I am too lazy to think about the sign.) The key point here is that the boundary term vanishes if either $\iota^\ast\alpha=0$ or $\iota^\ast\star\beta=0$, giving the two different boundary conditions. They are called “Dirichlet” and “Neumann” only because the first condition says something only about how $\alpha$ acts on tangential vectors, while the latter condition includes information about the normal component of $\beta$.
I am not sure how to answer your second question except to point at the Hodge Decomposition Theorem in your reference. I will point out that you have the wrong inclusion: Since $\iota^\ast d\omega = d\iota^\ast\omega$, we see that $\iota^\ast\omega=0$ implies $d\iota^\ast\omega=0$.