# zbMATH — the first resource for mathematics

Cohomology relative to a system of closed forms on the torus. (English) Zbl 1368.58011
Consider on the torus $$\mathbb T^{m+n}\doteq \mathbb T^m_t\times \mathbb T^n_x$$, $$t=(t_1,\dots,t_m)$$, $$x=(x_1,\dots, x_n)$$, a differential complex induced by the operator $\mathbb L_\Omega=d_t+\sum_{j=1}^n \omega^j\wedge \frac{\partial}{\partial x_j},$ where $$\omega^1,\dots, \omega^n$$ are real-valued, closed, one-forms on $$\mathbb T^m_t,$$ and $$d_t$$ is the exterior derivative.
Let $$A=(a_{kj})$$ denote the $$m\times n$$ matrix given by the periods of the one-forms above, that is, $$a_{kj}=\int_{\gamma_k}\omega^j$$, where $$\gamma_1, \dots , \gamma_m$$ are loops in $$\mathbb T^m$$ such that their homotopy classes form a basis of the first homology group $$H_1(\mathbb T^m).$$
It is assumed that $$A$$ satisfies the following Diophantine condition: there exist $$C>0$$ and $$\rho>0$$ such $\max\{|k_\ell+a_\ell\cdot J|, 1\leq \ell\leq m\}\geq C(|K|+|J|)^{-\rho},$ for all tuples of integers $$K\in\mathbb T^m$$ and $$J\in\mathbb T^n$$ such that $$K+AJ\not=0,$$ where $$a_\ell$$ is the $$\ell$$-th row of $$A$$ and $$k_\ell$$ is the $$\ell$$-th coordinate of $$K$$.
Let $$N=N(A)$$ be the subgroup of $$\mathbb Z^m$$ given by $$A(\mathbb Z^n)\cap \mathbb Z^m$$. Let $$\{h_1,\dots, h_r\}\subset \mathbb Z^n$$ be a set of generators of $$N$$, where $$r$$ is the rank of $$N$$.
Let $$\Psi:\mathbb T^m\rightarrow \mathbb T^r$$ be the linear map given by the matriz $$(AH)^T$$, where $$H$$ is the $$n\times r$$ matrix having $$h_1,\dots, h_r$$ as its columns.
The main result of this nice paper is that the $$p$$-th cohomology group relative to $$\mathbb L_\Omega$$, denoted by $$H^p_\Omega(\mathbb T^{m+n})$$, is isomorphic to $$H^p(\mathbb T^m)\otimes_\Psi C^\infty(\mathbb T^r)$$, the space generated by the forms $$\alpha\otimes \eta$$ with $$\alpha \in \Psi^*(C^\infty(\mathbb T^r))$$ and $$\eta\in H^p(\mathbb T^m)$$, the $$p$$-th De-Rham cohomology group of $$\mathbb T^m$$.

##### MSC:
 58J10 Differential complexes 35F35 Systems of linear first-order PDEs
Full Text:
##### References:
 [1] Bergamasco, Existence and regularity of periodic solutions to certain first-order partial differential equations, J. Fourier Anal. Appl. · Zbl 1357.35020 · doi:10.1007/s00041-016-9463-0 [2] Bergamasco, Global solvability of a class of involutive systems, J. Math. Anal. Appl. 233 (1) pp 314– (1999) · Zbl 0942.35011 · doi:10.1006/jmaa.1999.6310 [3] Bergamasco, Global solvability for certain classes of underdetermined systems of vector fields, Math. Z. 223 (2) pp 261– (1996) · Zbl 0863.58062 · doi:10.1007/PL00004558 [4] Bergamasco, Globally hypoelliptic systems of vector fields, J. Funct. Anal. 114 (2) pp 267– (1993) · Zbl 0777.58041 · doi:10.1006/jfan.1993.1068 [5] Berhanu, New Mathematical Monographs, 6 (2008) [6] Chen, Hypoelliptic vector fields and almost periodic motions on the torus Tn, Comm. Partial Differential Equations 25 (1-2) pp 337– (2000) · Zbl 0945.35007 · doi:10.1080/03605300008821516 [7] Greenfield, Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc. 31 pp 112– (1972) · doi:10.1090/S0002-9939-1972-0296508-5 [8] Hounie, Globally hypoelliptic vector fields on compact surfaces, Comm. Partial Differential Equations 7 (4) pp 343– (1982) · Zbl 0588.35064 · doi:10.1080/03605308208820226 [9] Meziani, Hypoellipticity of nonsingular closed 1-forms on compact manifolds, Comm. Partial Differential Equations 27 (7-8) pp 1255– (2002) · Zbl 1017.58014 · doi:10.1081/PDE-10486448 [10] Petronilho, Global hypoellipticity, global solvability and normal form for a class of real vector fields on a torus and application, Trans. Amer. Math. Soc. 363 (12) pp 6337– (2011) · Zbl 1235.35010 · doi:10.1090/S0002-9947-2011-05359-6 [11] Treves, Topological Vector Spaces, Distributions and Kernels (1967)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.