Differentiable Manifolds
In this chapter, we introduce the reader to the theory of manifolds. We start with the very notion of a manifold, illustrate it by a number of examples and discuss level sets in some detail. Thereafter, we carry over the concepts of differentiable mapping
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Differentiable Manifolds
In this chapter, we introduce the basic notions of the theory of manifolds. We start with the very notion of a manifold, illustrate it by a number of examples and discuss the class of examples provided by level sets in some detail. Next, we carry over the notions of differentiability, the concept of tangent space and the notion of derivative of a mapping from classical calculus to the theory of manifolds. In this context, we also generalize the basic theorems of classical calculus to the case of manifolds. Finally, we discuss some more advanced topics needed later on: the notion of submanifold and the concept of transversality. Since the notion of a submanifold is quite subtle, we treat this subject in detail. In our terminology, a submanifold is defined by an injective immersion. There are two important special classes of submanifolds showing up in various applications. They are called embedded and initial, respectively. Throughout the text, the reader will find a large number of illustrative examples.
1.1 Basic Notions and Examples Manifolds are topological spaces which locally look like Rn . Therefore, they allow for an extension of the notions of classical calculus. For the topological notions used in this section we refer the reader to the standard literature, e.g., [53], [55], [199] or [267]. In the sequel, we will use the following notation. An element x ∈ Rn is an n-tuple written as x = (x1 , . . . , xn ). The Euclidean scalar product on Rn is denoted by x·y=
n
xi yi
i=1
and the corresponding norm is denoted by · . For the standard basis in Rn we write {e1 , . . . , en }. The dual basis {e∗1 , . . . , e∗n } in Rn∗ is defined by e∗i (ej ) = δ i j . For x ∈ Rn , its coefficients in the standard basis are given by x i = e∗i (x). G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_1, © Springer Science+Business Media Dordrecht 2013
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1 Differentiable Manifolds
Thus, we have x = ni=1 x i ei . Of course, numerically, the numbers x i and xi coincide. If there is no danger of confusion, we will use the Einstein summation convention, that is, we will also write x = x i ei . Definition 1.1.1 (Topological manifold) A topological space M is called a topological manifold if it is Hausdorff, second countable and locally homeomorphic to Rn for some fixed n ∈ N. This means that for every m ∈ M, there exists an open neighbourhood U of m in M and a mapping κ : U → Rn such that κ(U ) is open in Rn and κ is a homeomorphism onto its image. The pair (U, κ) is called a local chart for M. A family A = {(Uα , κα ) : α ∈ A} of local charts satisfying α∈A Uα = M is called an atlas for M. The number n is called the dimension of M. Let (U, κ) be a local chart on M. If m ∈ U , we say that (U, κ) is a chart at m. The functions κ i := e∗i ◦ κ : U → R,
1 ≤ i ≤ n,
define a system of local coordinates on U . The numbers κ i (m) are called the local coordinates of m in the chart (U, κ). In particular, (Rn ,
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