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For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold.

Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.

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There certainly isn't a simple general statement like the Hodge decomposition in any category that contains, for example, complete noncompact manifolds with some pointwise condition on curvature, even if they are diffeomorphic to products of a compact factor and a real line. But you could try MR1815415 (2002j:58033) Ahmed, Zulfikar M.(1-CLMB); Stroock, Daniel W.(1-MIT) A Hodge theory for some non-compact manifolds. J. Differential Geom. 54 (2000), no. 1, 177–225. 58J05 (58A14 58J65)

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  • $\begingroup$ You might also want to look at K. Kodaira, Harmonic fields in Riemannian manifolds (generalized potential theory), Ann. Math. 50 (1949) 587– 665. and M. Gromov, Kähler hyperbolicity and $L^2$-Hodge theory, J. Differential Geom. 33 (1991) 253–320, $\endgroup$
    – Ben McKay
    Jul 24 '16 at 10:08
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Perhaps it will be useful to know that in that case there is an isomorphism of the cohomology groups with compact support: $$ \pi_* : H^*_c(M\times\mathbb{R})\to H^{*-1}_c(M). $$ This is Proposition 4.7 in R. Bott, L. W. Tu, Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982.

Regarding the Hodge decomposition we have the following result in Sobolev spaces.

Theorem ($L^p$-Hodge Decomposition). Let $M$ be a smooth, compact, $k$-dimensional oriented manifold without boundary and let $\Omega\subset M$ be an open subset. Then for any $p\in (1,\infty)$ and any $\ell$-form $\eta\in L^p(\bigwedge^\ell\Omega)$, $1\leq \ell\leq k$ there exist $\omega_1\in W^{1,p}(\bigwedge^{\ell-1}\Omega)$, $\omega_2\in W^{1,p}(\bigwedge^{\ell+1}\Omega)$ such that $$ \eta=d\omega_1+\delta\omega_2+h $$ where $h\in C^\infty(\bigwedge^\ell\Omega)$ is closed $dh=0$ and co-closed $\delta h=0$ and hence harmonic.

I believe that if $\eta\in C^\infty$, then $\omega_1,\omega_2\in C^\infty$ so the classical case should be covered.

In the case when $\Omega=M$ the theorem is Proposition 6.5 in

C. Scott, $L^p$ theory of differential forms on manifolds. Trans. Amer. Math. Soc., 347 (1995), 2075-2096.

In the case of a general open set we simply extend $\eta$ to $L^p(\bigwedge^\ell M)$ by zero, apply the Hodge decomposition on $M$ and restrict all the resulting forms to $\Omega$.

Note that the above result applies to the manifold $M\times (0,1)$ since it can be isometrically embedded into $M\times S^1$ as an open set.

I know that this is not exactly the answer to the question, but I hope it could be helpful.

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