# a non orientable 3d-manifold with boundary

Mö x I

The picture on the right is representation and a description of its parts of  a 3d chunk, it is the trivial $I$-bundle over the möbius-strip, $M\ddot{o}\times I$. It is  useful to determine which 3d spaces are non orientable.

A 3d-space is non orientable if it has a simple closed curve whose 3d regular neighborhood is homeomorphic to the model of the picture.

If we denote by $C\ddot{o}$ the core of the möbius strip, you can deduce for the curve $C\ddot{o}\times\{\frac{1}{2}\}$ (the black pearled one) that its tangent bundle can’t be embedded in the tangent bundle of $\mathbb{R}^3$ or in any other tangent bundle of an orientable one.

Remember $N_2$ is the Klein bottle. Let’s the image talks by itself.

Ah, il nome di questo grande pezzo è la bottiglia solida di Klein.