Preliminaries of Differentiable Manifolds

Before introducing the concept of differentiable manifold, we first explain what mapping is. Given two sets X, Y, and a corresponding principle, if for any x ∊ X, there exists y = f(x) ∊ Y to be its correspondence, then f is a mapping of the set X into the set Y, which is denoted as f : X → Y. X is said to be the domain of definition of f, and f(x) = {f(x) | x ∊ X} ⊂ Y is said to be the image of f. If f(X) = Y, then f is said to be surjective or onto; if f(x) = f(x′) ⇒ x = x′, then f is said to be injective (one-to-one); if f is both surjective and injective (i.e., X and Y have a one-to-one correspondence under f), f is said to be bijective. For a bijective mapping f, if we define x = f −1(y), then f −1 : Y → X is said to be the inverse mapping of f. In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). For example, for two groups G and G′ and a mapping f : G → G′, a → f(a), if f(a, b) = f(a) · f(b), ∀a, b ∊ G, then f is said to be a homomorphism from G to G′. A homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structures, i.e., properties such as identity element, inverse element, and binary operations. An isomorphism is a bijective homomorphism. If f is a G → G′ homomorphic mapping, and also a one-to-one mapping from G to G′, then f is said to be a G → G′ isomorphic mapping. An epimorphism is a surjective homomorphism. Given two topological spaces (x, τ) and (y, τ), if the mapping f : X → Y is one-to-one, and both f and its inverse mapping f −1 : Y → X are continuous, then f is said to be a homeomorphism. If f and f −1 are also differentiable, then the mapping is said to be diffeomorphism. A monomorphism (sometimes called an extension) is an injective homomorphism. A homomorphism from an object to itself is said to be an endomorphism. An endomorphism that is also an isomorphism is said to be an automorphism. Given two manifolds M and N, a bijective mapping f from M to N is called a diffeomorphism if both f : M → N and its inverse f −1 : N → M are differentiable (if these functions are r times continuously differentiable, f is said to be a Cr-diffeomorphism).