Non orientable manifold.

Question

I'm looking for a proof of the following statement:

Let $M$ be a smooth manifold covered by two connected charts $(U,\phi) , (V,\psi)$ such that $U\cap V$ has exactly two connected components, with the following propertie: determinant of change of coorinates is positive in one and negative in the other. Then $M$ is not orientable.

@DietrichBurde sorry, I need to prove that M is not orientable. — Peter Languilla, Jun 20 '19 at 14:34
That's not a question, just a statement of (presumably) fact. Were you meaning to ask a question about a particular problem you've encountered while conducting that proof? — hmakholm left over Monica, Jun 20 '19 at 14:48
What is your definition of orientability, please? That a certain obstruction vanishes? That $w_1 = 0$? That there's a set of charts with a certain property? — John Hughes, Jun 20 '19 at 14:54
@JohnHughes my definition of orientability is: $M$ is orientable if exists an atlas with change of coordinates all positives. — Peter Languilla, Jun 20 '19 at 14:57
@ArcticChar: Not a duplicate of that (without almost as much knowledge as you need to provide a complete answer). But certainly closely related ... by the answer to that question. — John Hughes, Jun 20 '19 at 19:03

score 1 · Answer 1 · answered Jun 20 '19 at 16:53

1

See Andrew Hwang's answer to this question: Manifold is not orientable

That's really all you need.

(The question isn't very well phrased, but the answer is still the tool you need!)

answered Jun 20 '19 at 16:53

John Hughes

93,729

Non orientable manifold.

1 Answers1