Integration on Manifolds Part 1

Problem 3.8 Prove: (i) The product of two orientable manifolds is orientable. (ii) The total space of the tangent bundle over any manifold is an orientable manifold.

Problem 3.9 Prove that if a $C^{\infty}$ manifold $M$ admits an atlas formed by two charts $(U, \varphi),(V, \psi)$, and $U \cap V$ is connected, then $M$ is orientable. Apply this result to the sphere $S^{n}, n>1$, with the atlas formed by the stereographic projections from the poles (see Problem 1.28).

Problem 3.10 Study the orientability of the following $C^{\infty}$ manifolds: (i) A cylindrical surface of $\mathbb{R}^{3}$, with the atlas given in Problem 1.30. (ii) The Möbius strip, with the atlas given in Problem 1.31. (iii) The real projective space $\mathbb{R} P^{2}$, with the atlas given in Problem 1.81.