Reminders from differential geometry

This book assumes that the reader is familiar with basic notions of differential geometry, as covered for example, in. For reader's convenience, in this section we briefly remind some definitions and fix notation.

Unless otherwise specified, all manifolds considered in this book will be C∞ real manifolds; the word “smooth” will mean C∞. All manifolds we will consider will have at most countably many connected components.

For a manifold M and a point m ∈ M, we denote by TmM the tangent space to M at point m, and by TM the tangent bundle to M. The space of vector fields on M (i.e., global sections of TM) is denoted by Vect(M). For a morphism f:X → Y and a point x ∈ X, we denote by f*:TxX → Tf(x)Y the corresponding map of tangent spaces.

Recall that a morphism f:X → Y is called an immersion if rank f* = dimX for every point x ∈ X; in this case, one can choose local coordinates in a neighborhood of x ∈ X and in a neighborhood of f(x) ∈ Y such that f is given by f (x1, … xn) = (x1, …, xn, 0, … 0).