Can $\mathbb{R}\mathbb{P}^2$ be embedded into an orientable 3-manifold?

Question

We know that $\mathbb{R}\mathbb{P}^2$ cannot be embedded into $\mathbb{R}^3$, but is there an orientable 3-manifold where it is possible?

Answer 1

Yes, the obvious vector space inclusion $\iota: \mathbb{R}^3 \hookrightarrow \mathbb{R}^4$, or its restriction $\iota\vert_{\mathbb{S}^2} : \mathbb{S}^2 \hookrightarrow \mathbb{S}^3$, induces an embedding of $\mathbb{RP}^2$ into $\mathbb{RP}^3$, and the latter is orientable.