Leonor Godinho - An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity

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Springer International Publishing Switzerland 2014

Leonor Godinho and Jos Natrio An Introduction to Riemannian Geometry Universitext 10.1007/978-3-319-08666-8_1

1. Differentiable Manifolds

Leonor Godinho 1

(1)

Departamento de Matemtica, Instituto Superior Tcnico, Lisbon, Portugal

Leonor Godinho (Corresponding author)

Email:

Jos Natrio

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In pure and applied mathematics, one often encounters spaces that locally look like , in the sense that they can be locally parameterized by coordinates: for example, the -dimensional sphere , or the set of configurations of a rigid body. It may be expected that the basic tools of calculus can still be used in such spaces; however, since there is, in general, no canonical choice of local coordinates, special care must be taken when discussing concepts such as derivatives or integrals whose definitions in rely on the preferred Cartesian coordinates.

The precise definition of these spaces, called differentiable manifolds , and the associated notions of differentiation, are the subject of this chapter. Although the intuitive idea seems simple enough, and in fact dates back to Gauss and Riemann, the formal definition was not given until 1936 (by Whitney).

The concept of spaces that locally look like is formalized by the definition of topological manifolds : topological spaces that are locally homeomorphic to . These are studied in Sect. , where several examples are discussed, particularly in dimension (surfaces).

Differentiable manifolds are defined in Sect..

Vector fields and their flows are the main topic of Sect.. A natural differential operation between vector fields, called the Lie bracket , is defined; it measures the non-commutativity of their flows and plays a central role in differential geometry.

Section is devoted to the important class of differentiable manifolds which are also groups, the so-called Lie groups . It is shown that to each Lie group one can associate a Lie algebra , i.e. a vector space equipped with a Lie bracket. Quotients of manifolds by actions of Lie groups are also treated.

Orientability of a manifold (closely related to the intuitive notion of a surface having two sides) and manifolds with boundary (generalizing the concept of a surface bounded by a closed curve, or a volume bounded by a closed surface) are studied in Sects..

1.1 Topological Manifolds

We will begin this section by studying spaces that are locally like , meaning that there exists a neighborhood around each point which is homeomorphic to an open subset of .

Definition 1.1

A topological manifold of dimension is a topological space with the following properties:

(i)

is Hausdorff , that is, for each pair of distinct points of there exist neighborhoods of and such that .

(ii)

Each point possesses a neighborhood homeomorphic to an open subset of .

(iii)

satisfies the second countability axiom , that is, has a countable basis for its topology.

Conditions (i) and (iii) are included in the definition to prevent the topology of these spaces from being too strange. In particular, the Hausdorff axiom ensures that the limit of a convergent sequence is unique. This, along with the second countability axiom, guarantees the existence of partitions of unity (cf. Sect. 7.2 of Chap. ), which, as we will see, are a fundamental tool in differential geometry.

Remark 1.2

If the dimension of is zero then is a countable set equipped with the discrete topology (every subset of is an open set). If , then is locally homeomorphic to an open interval; if , then it is locally homeomorphic to an open disk etc.

Example 1.3

(1)

Every open subset

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