# Existence of certain surfaces in flat riemannian 3-manifold

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.

• Thurston's geometrization tells you that $M$ should be mapping torus of finite automorphism group of $T^2$. So you can always find an incompressible torus there which has positive genus – Anubhav Mukherjee Nov 6 '18 at 18:09
• @AnubhavMukherjee Can you point me any reference? – Michael Nov 7 '18 at 12:26