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It is well known that a bundle which admits a compatible foliation* supports a flat connection and the converse is also true (every smooth and finite dimensional). The question is:

Does every bundle with compact total space over the n-dimensional torus admit a compatible foliation?

  • that is, for every point in the base, there exists a local trivialization around this point such that the horizontal leaves are plaques of the foliation.
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Since the connected component of the diffeomorphism group of the torus is homotopic equivalent to the torus a fibre space over the torus whose typical fibre is the torus is equivalent to a principal $\mathbb{T} ^2$-bundle. Every element of $H^2(\mathbb{T}^n,\mathbb{Z})$ is the Chern class of a $S^1$-bundle over $\mathbb{T}^n$ which is not flat if the Chern class does not vanish.

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  • $\begingroup$ Does are orientable the total space of those bundles? In case of higher dimensional fiber, does are analogous characterization? Thanks. $\endgroup$ – Elizeu França May 29 '18 at 11:21
  • $\begingroup$ Yes, they are orientable , the same method can be used to construct a $\mathbb{T}^m$ bundle over $\mathbb{T}^n$ by considering $H^2(\mathbb{T}^n,\mathbb{Z}^m)$. $\endgroup$ – Tsemo Aristide May 29 '18 at 11:27
  • $\begingroup$ Do you have a reference to indicate to me? Thanks. $\endgroup$ – Elizeu França May 29 '18 at 12:47
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    $\begingroup$ Let $\omega$ be a closed two form, consider a good cover $(U_i)$, you have $\omega_{\mid U_i}=d\alpha_i$, $(\alpha_j-\alpha_i)_{\mid U_i\cap U_j}$ is closed it is equal to a function $f_{ij}:U_i\cap U_j\rightarrow\mathbb{R}$. Since $\omega\in H^2(\mathbb{T}^n,\mathbb{Z})$, $f_{ij}-f_{ik}+f_{jk}\in \mathbb{Z}$, $g_{ij}=exp(if_{ij})$ define the localization of the bundle. $\endgroup$ – Tsemo Aristide May 29 '18 at 13:37
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    $\begingroup$ In general they would not be diffeomorphic. For instance, take n=2. Then it's not hard to see that the fundamental group of a circle bundle with non-trivial Euler class is not abelian, whereas it is abelian for the trivial bundle. $\endgroup$ – Danny Ruberman May 29 '18 at 20:49

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