The picture on the right is representation and a description of its parts of a 3d chunk, it is the trivial -bundle over the möbius-strip, . 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 the core of the möbius strip, you can deduce for the curve (the black pearled one) that its tangent bundle can’t be embedded in the tangent bundle of or in any other tangent bundle of an orientable one.
Remember is the Klein bottle. Let’s the image talks by itself.
Ah, il nome di questo grande pezzo è la bottiglia solida di Klein.