Abstract
The problems treated in the present chapter are concerned with orientability of smooth manifolds, especially the orientability of several manifolds introduced in the previous chapter (such as the cylindrical surface, the Möbius strip, and the real projective space ℝ2) is considered.
Some attention is paid to integration on chains and integration on oriented manifolds, by applying Stokes’ and Green’s Theorems. Some calculations of de Rham cohomology are proposed, such as the cohomology groups of the circle and of an annular region in the plane. This cohomology is also used to prove that the torus T2 and the sphere S2 are not homeomorphic.
The chapter ends with an application of Stokes’ Theorem to certain structure on the complex projective space ℂP n.
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© 2009 Springer Science+Business Media B.V.
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Gadea, P.M., Muñoz Masqué, J. (2009). Integration on Manifolds. In: Analysis and Algebra on Differentiable Manifolds. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3564-6_3
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DOI: https://doi.org/10.1007/978-90-481-3564-6_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3563-9
Online ISBN: 978-90-481-3564-6
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