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# Manual Manifolds and differential forms

Such a function has an integral in the usual Riemann or Lebesgue sense. Fixing an orientation is necessary for this to be well-defined. Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined. The orientation resolves this ambiguity. First, assume that there is a parametrization of M by an open subset of Euclidean space. That is, assume that there exists a diffeomorphism.

In coordinates, this has the following expression. Fix a chart on M with coordinates x 1 , In general, an n -manifold cannot be parametrized by an open subset of R n.

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But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations. To make this precise, it is convenient to fix a standard domain D in R k , usually a cube or a simplex.

That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a k -dimensional submanifold of M. If the chain is. This approach to defining integration does not assign a direct meaning to integration over the whole manifold M. However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly triangulated in an essentially unique way, and the integral over M may be defined to be the integral over the chain determined by a triangulation.

There is another approach, expounded in Dieudonne , which does directly assign a meaning to integration over M , but this approach requires fixing an orientation of M. On this chart, it may be pulled back to an n -form on an open subset of R n. Here, the form has a well-defined Riemann or Lebesgue integral as before. It is also possible to integrate k -forms on oriented k -dimensional submanifolds using this more intrinsic approach. The form is pulled back to the submanifold, where the integral is defined using charts as before.

Fubini's theorem states that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product.

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7. This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well. The geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well.

Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors. Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers.

Let M and N be two orientable manifolds of pure dimensions m and n , respectively. Suppose that. Following Dieudonne , there is a unique. That is, suppose that.

## Sasakian Manifolds: Differential Forms, Curvature and Conformal Killing Forms

In particular, a choice of orientation forms on M and N defines an orientation of every fiber of f. The analog of Fubini's theorem is as follows. Fix orientations of M and N , and give each fiber of f the induced orientation. Moreover, there is an integrable n -form on N defined by.

It is also possible to integrate forms of other degrees along the fibers of a submersion. Integration along fibers is important for the construction of Gysin maps in de Rham cohomology. Integration along fibers satisfies the projection formula Dieudonne This case is called the gradient theorem , and generalizes the fundamental theorem of calculus. This path independence is very useful in contour integration. This theorem also underlies the duality between de Rham cohomology and the homology of chains.

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On a general differentiable manifold without additional structure , differential forms cannot be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the 1 -form dx over the interval [0, 1].

By contrast, the integral of the measure dx on the interval is unambiguously 1 formally, the integral of the constant function 1 with respect to this measure is 1.

Similarly, under a change of coordinates a differential n -form changes by the Jacobian determinant J , while a measure changes by the absolute value of the Jacobian determinant, J , which further reflects the issue of orientation. In the presence of the additional data of an orientation , it is possible to integrate n -forms top-dimensional forms over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the fundamental class of the manifold, [ M ]. Formally, in the presence of an orientation, one may identify n -forms with densities on a manifold ; densities in turn define a measure, and thus can be integrated Folland , Section On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate n -forms over compact subsets, with the two choices differing by a sign.

On non-orientable manifold, n -forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere there are no volume forms on non-orientable manifolds , but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate n -forms.

One can instead identify densities with top-dimensional pseudoforms. Geometrically, a k -dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by degrees. Compare the Gram determinant of a set of k vectors in an n -dimensional space, which, unlike the determinant of n vectors, is always positive, corresponding to a squared number.

An orientation of a k -submanifold is therefore extra data not derivable from the ambient manifold. On a Riemannian manifold, one may define a k -dimensional Hausdorff measure for any k integer or real , which may be integrated over k -dimensional subsets of the manifold.

A function times this Hausdorff measure can then be integrated over k -dimensional subsets, providing a measure-theoretic analog to integration of k -forms. The n -dimensional Hausdorff measure yields a density, as above. The differential form analog of a distribution or generalized function is called a current. The space of k -currents on M is the dual space to an appropriate space of differential k -forms. Currents play the role of generalized domains of integration, similar to but even more flexible than chains.

Differential forms arise in some important physical contexts. For example, in Maxwell's theory of electromagnetism , the Faraday 2-form , or electromagnetic field strength , is.

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