# Does every fiber bundle admits flat bundle structure?

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.

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.
• Yes, they are orientable , the same method can be used to construct a $\mathbb{T}^m$ bundle over $\mathbb{T}^n$ by considering $H^2(\mathbb{T}^n,\mathbb{Z}^m)$. May 29 '18 at 11:27
• 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. May 29 '18 at 13:37