Graph of a smooth map is a smooth manifold but is it orientabl

Asked Dec 06 '22 at 19:59

Active Dec 06 '22 at 19:59

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I got to know that the Tangent Bundle of a smooth manifold is Orientable I read that the graph of a smooth map is a smooth manifold. Now I am thinking if it is always orientable or not.

I think it is not.

Define some map $f:U \to \mathbb{R}$ such that $$\Gamma_f=\{(x,y),f(x,y)) |(x,y)\in U \subset \mathbb{R}^2\}$$ gives us the Mobius strip, but I am not sure if I can find such a smooth function. But I feel it need not be a orientable manifold in general.

asked Dec 06 '22 at 19:59

permutation_matrix

1,356

2

The graph is diffeomorphic to the domain. Do you see why? Dose that answer the question? – Thorgott Dec 06 '22 at 20:03
yeah, and morever ws have a global cover sending $((x,y),f(x,y)) \to (x,y)$ and so the Jacobian is identity. So it must be orientable. – permutation_matrix Dec 06 '22 at 20:06
Maybe you want a more interesting domain and codomain? – Ted Shifrin Dec 06 '22 at 20:35
@TedShifrin, couldn't get you. – permutation_matrix Dec 06 '22 at 21:27

Graph of a smooth map is a smooth manifold but is it orientabl

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