Differentiable manifolds

Abstract

This chapter is devoted to propose problems on the basics of differentiable manifolds including—among others—the following topics: smooth mappings, critical points and critical values of mappings, immersions and submersions, construction of manifolds by inverse image, submersions and quotient manifolds, tangent bundles and vector fields, including integral curves and flows.

Functions and other objects are assumed to be of class C^∞ (also called indiscriminately as ‘differentiable’), mainly to simplify the approach. Similarly, manifolds are supposed to be Hausdorff and second countable, although a section is included analysing what happens when one of these properties fails to hold, aimed at a better understanding of the meaning of such properties. In this first chapter we have deliberately included many examples and figures.

As important and non-trivial instances of differentiable manifolds, the real projective space ℝPⁿ and the real Grassmannian G _k (ℝⁿ) are studied with some detail.