Integration on Manifolds

After giving some definitions and results on orientability of smooth manifolds, the problems treated in the present chapter are concerned with orientation of smooth manifolds; especially the orientation of several manifolds introduced in the previous chapter, such as the cylindrical surface, the Mobius strip, and the real projective space ℝP2. 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 T 2 and the sphere S 2 are not homeomorphic. The chapter ends with an application of Stokes’ Theorem to a certain structure on the complex projective space ℂP n .