Given a Riemannian manifold $(M, g)$ of dimension $n$, the Hodge star operator $\star: \Omega^k(M) \to \Omega^{n-k}(M)$ is defined. What is the (formal) adjoint of $\star$ under the integration pairing? That is, I'm looking for an operator $\sharp$ that assigns to a $(n-k)$-dimensional submanifold $S$ a $k$-dimensional (singular?) submanifold $\sharp S$ in such a way that $$ \int_S \star \, \alpha = \int_{\sharp S} \alpha $$ holds for all $\alpha \in \Omega^k(M)$.

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    $\begingroup$ https://en.wikipedia.org/wiki/Current_(mathematics) $\endgroup$ – abx Dec 26 '18 at 13:49
  • $\begingroup$ The answer is built-in your question. $\endgroup$ – Liviu Nicolaescu Dec 26 '18 at 14:32
  • $\begingroup$ Sure, I can always view the last equality as a defining equation for a current. My question is rather if this current has a geometric interpretation or can be constructed by other means. My inspiration came from the fact that the exterior differential has the boundary operator as the adjoint and I was hoping for something comparable for the Hodge operator, perhaps using some kind of flow constructed with the help of the Riemannian metric. $\endgroup$ – Tobias Diez Dec 26 '18 at 23:13

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