Geodesic
Homogenization of harmonic 1-forms on pseudo-Riemannian manifolds of complicated microstructure
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004) no. 2, pp. 249-257 
A. P. Rybalko. Homogenization of harmonic 1-forms on pseudo-Riemannian manifolds of complicated microstructure. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004) no. 2, pp. 249-257. http://geodesic.mathdoc.fr/item/JMAG_2004_11_2_a9/
@article{JMAG_2004_11_2_a9,
     author = {A. P. Rybalko},
     title = {Homogenization of harmonic 1-forms on {pseudo-Riemannian} manifolds of complicated microstructure},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {249--257},
     year = {2004},
     volume = {11},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2004_11_2_a9/}
}
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4-dimentional manifolds $\tilde M_\varepsilon^4=\mathbf R\times M_\varepsilon^3$, where $M_\varepsilon^3$ are Riemannian manifolds of complicated microstructure are considered. $M_\varepsilon^3$ consist of two copies of $\mathbf R^3$ with a large number of holes connected in pairs by means of fine tubes. The asymptotic behaviour of harmonic $1$-forms on $\tilde M_\varepsilon^4$ is studied as $\varepsilon\to 0$, when the number of tubes on $M_\varepsilon^3$ tends to infinity and their radii tend to zero. The homogenized equations on $\mathbf R^4$ describing the leading term of the asymptotics are obtained. The result of homogenization of the solution of Cauchy problem for wave equation on $\tilde M_\varepsilon^4$ as $\varepsilon\to 0$ is obtained.