I am having some trouble understanding the notion of an orientated manifold. But first let me get some preliminary definitions out of the way:(adsbygoogle = window.adsbygoogle || []).push({});

A diffeomorphism is said to be orientation-preserving if the determinant of its Jacobian is positive. A k-manifold M in [itex] \mathbb{R}^n [/itex] is said to be orientable if there is an atlas of coordinate patches, [itex] \vartheta = \{\alpha_i : U_i \rightarrow V_i \} [/itex] covering M such that the transition functions are orientation-preserving. This atlas is said to be an orientation on M.

I am trying to geometrically visualize this. My book gives some help in the case where k = 1, k = n, or k = n - 1.

Let us suppose for the moment that k = n - 1, because that is what I am going to do most of my visualizing in. And let us use in our example the 2-sphere, [itex] S^2 [/itex], which is a 2-manifold in [itex] \mathbb{R}^3 [/itex].

Let's picture the unit normal field to [itex] S^2 [/itex], corresponding to some orientation that we give [itex] S^2 [/itex]. Now, because [itex] S^2 [/itex] is orientable, we can picture the unit normal field corresponding to the given orientation, as say, all of the normal vectors to the sphere that are pointing outwards. The only other possible unit normal field to [itex] S^2 [/itex] is the one where all of the vectors which are normal to the sphere point inward, toward the origin.

Now, tell me if my reasoning is correct here. The ONLY reason that we can find a unit normal field to [itex] S^2 [/itex] such that all of the normal vectors point in the same direction (i.e. outward in this example) is because [itex] S^2 [/itex] is orientable. If we had another manifold, say the Mobius Strip, then we cannot find a unit normal field such that each normal vector is pointing outwards, because as we travel around the strip, there will be normal vectors starting to point in the opposite direction. So because of this, the Mobius strip is not orientable.

Is my reasoning correct here? And also, why do we care if a manifold is orientable or not? What purpose does it serve? And suppose that we do indeed have an orientable manifold, M, with two possible choices of orientation. How does the choice of orientation affect the structure of the manifold?

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# Understanding Orientated Manifolds

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