Integration on Manifolds

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 ℂPⁿ.