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I have been trying to self-learn differential geometry. I think I may have misunderstood/missed out on something along the way.

It is said that for $X$ an $n$-form, $M$ a differentiable manifold, $\phi:M\to M$ a diffeomorphism that is orientation preserving. Then $$\int_M \phi^* (X)=\int_M X$$ As my understanding goes, 2 orientations $\epsilon,\epsilon'$ are equivalent if $\epsilon=f\epsilon'$ where $f$ is an everywhere positive function.

But I don't understand how the above works. Also, is there an intuitive explanation?

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It is nothing but "change of variables" or "integration by substitution". – Xipan Xiao Jul 29 '13 at 17:52
up vote 3 down vote accepted

Using differential forms to define orientations is not always the clearest way to do things. A more friendly definition of orientation can be built by first defining the orientation of a vector space. An orientation of a vector space is an ordering of the basis, up to an even permutation. In two-dimensional space, that comes down to the difference between clockwise and counterclockwise, and in three-dimensional space, the difference between right-handed and left-handed systems.

Attached to a smooth manifold $M$ is a collection of tangent spaces $T_{p}M$ at each point, and each such tangent space is a vector space. If $M$ was a sphere, and $T_{p}M$ a tangent plane, a choice of orientation is simultaneously a choice of "in" and "out", as well as a choice of "clockwise" and "counterclockwise". To see that these are the same things, just curl your fingers and see where to put your thumb, just as in your elementary physics/calculus classes.

An orientation on a manifold is a consistent way of choosing "in" or "out" for each tangent space. In concrete terms, it is an atlas for $M$ where the Jacobians of the transition functions always have positive determinant. Intuitively, as one jumps from one coordinate patch to the next, we should have agreement as to the orientation of the tangent space.

A diffeomorphism always has invertible Jacobian, and so the determinant of the Jacobian has constant sign (positive or negative) on a connected manifold. An orientation-preserving diffeomorphism is one that has everywhere positive Jacobian determinant. Two orientations are deemed equivalent if there is an orientation-preserving diffeomorphism between them, and using this definition, there are only two possible orientations of a smooth manifold, if it can be oriented at all. This is not a trivial assertion, since there exists manifolds (e.g. the Möbius band) that cannot admit an atlas of everywhere positive Jacobian determinant.

To jump back to differential forms, smooth manifold have somethings called "top forms" that allow for integration on the manifold. One can think of them as defining some notion of infinitesimal density and direction. It turns out that the existence of a non-vanishing top form (called a "volume form") is equivalent to being orientable. Roughly speaking, this is because a volume form looks locally like Euclidean volume, and the atlas maps that witness this have positive Jacobian determinant. Another way of thinking about it is that a volume form acts on a basis of vectors, and can decide on a choice of basis of $T_{p}M$ that gives rise to an orientation. To sketch this idea more summarily, a volume form is a system of density and direction that can decide which directions should give positive integral, and hence decides some notion of "up".

The integral equality you have there encapsulates the above explanation. Given a diffeomorphism of $M$, we can push around a volume form to create a new volume form, as specified by the nature of the diffeomorphism and its derivative. The integral over the whole manifold is either the same, indicating that the diffeomorphism kept the direction of the volume form (i.e. the diffeomorphism had everywhere positive Jacobian determinant), or the the integral is negative, because we flipped the volume form upside down. As we can see, the former case coincides exactly with how we defined an orientation-preserving diffeomorphism.

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Good answer, but one nitpick: "there are only two possible orientations of a smooth manifold, if it can be oriented at all" is only true for connected manifolds (of course the interesting case for orientability, but should probably made explicit nevertheless.) – Daniel Fischer Jul 29 '13 at 18:19
Thank you, Issac! This is a very good explanation! – explorer Jul 29 '13 at 18:55
@DanielFischer: Thanks for the clarification! – explorer Jul 29 '13 at 18:55

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