Let $(M,g)$ be a closed orientable connected riemannian $3$-manifold (closed means compact and without boundary).
Is the following affirmation true?
Afirmação: If $(M,g)$ is flat, then there is a closed orientable connected surface $\Sigma$ embedded in $M$ such that $\Sigma$ has positive genus and the induced homomorphism in the fundamental groups by inclusion $i:\Sigma\rightarrow M$ is injective.