An Introduction to Riemannian Geometry

With Applications to Mechanics and Relativity

verfasst von: Leonor Godinho, José Natário

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity.

The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects.

The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Differentiable Manifolds

Abstract

In pure and applied mathematics, one often encounters spaces that locally look like \(\mathbb {R}^n\), in the sense that they can be locally parameterized by \(n\) coordinates: for example, the \(n\)-dimensional sphere \(S^n \subset \mathbb {R}^{n+1}\), or the set \(\mathbb {R}^3 \times SO(3)\) 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 \(\mathbb {R}^n\) 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.

Leonor Godinho, José Natário

Chapter 2. Differential Forms

Abstract

This chapter discusses integration on differentiable manifolds. Because there is no canonical choice of local coordinates, there is no natural notion of volume, and so only objects with appropriate transformation properties under coordinate changes can be integrated. These objects, called differential forms, were introduced by Élie Cartan in 1899; they come equipped with natural algebraic and differential operations, making them a fundamental tool of differential geometry.

Leonor Godinho, José Natário

Chapter 3. Riemannian Manifolds

Abstract

The metric properties of \(\mathbb {R}^n\) (distances and angles) are determined by the canonical Cartesian coordinates. In a general differentiable manifold, however, there are no such preferred coordinates; to define distances and angles one must add more structure by choosing a special \(2\)-tensor field, called a Riemannian metric (much in the same way as a volume form must be selected to determine a notion of volume). This idea was introduced by Riemann in his 1854 habilitation lecture “On the hypotheses which underlie geometry”, following the discovery (around 1830) of non-Euclidean geometry by Gauss, Bolyai and Lobachevsky (in fact, it was Gauss who suggested the subject of Riemann’s lecture). It proved to be an extremely fruitful concept, having led, among other things, to the development of Einstein’s general theory of relativity. This chapter initiates the study of Riemannian geometry.

Leonor Godinho, José Natário

Chapter 4. Curvature

Abstract

The local geometry of a general Riemannian manifold differs from the flat geometry of the Euclidean space \(\mathbb {R}^n\): for example, the internal angles of a geodesic triangle in the \(2\)-sphere \(S^2\) (with the standard metric) always add up to more than \(\pi \). A measure of this difference is provided by the notion of curvature, introduced by Gauss in his 1827 paper “General investigations of curved surfaces”, and generalized to arbitrary Riemannian manifolds by Riemann himself (in 1854). It can appear under many guises: the rate of deviation of geodesics, the degree of non-commutativity of covariant derivatives along different vector fields, the difference between the sum of the internal angles of a geodesic triangle and \(\pi \), or the angle by which a vector is rotated when parallel-transported along a closed curve. This chapter addresses the various characterizations and properties of curvature.

Leonor Godinho, José Natário

Chapter 5. Geometric Mechanics

Abstract

Mechanics, the science of motion, was basically started by Galileo and his revolutionary empirical approach. The first precise mathematical formulation was laid down by Newton in the Philosophiae Naturalis Principia Mathematica, first published in 1687, which contained, among many other things, an explanation for the elliptical orbits of the planets around the Sun. Newton’s ideas were developed and extended by a number of mathematicians, including Euler, Lagrange, Laplace, Jacobi, Poisson and Hamilton. Celestial mechanics, in particular, reached an exquisite level of precision: the 1846 discovery of planet Neptune, for instance, was triggered by the need to explain a mismatch between the observed orbit of planet Uranus and its theoretical prediction. This chapter uses Riemannian geometry to give a geometric formulation of Newtonian mechanics.

Leonor Godinho, José Natário

Chapter 6. Relativity

Abstract

This chapter studies one of the most important applications of Riemannian geometry: the theory of general relativity. This theory, which ultimately superseded the classical mechanics of Galileo and Newton, arose from the seemingly paradoxical experimental fact that the speed of light is the same for every observer, independently of their state of motion. In 1905, after a period of great confusion, Einstein came up with an explanation that was as simple as it was radical: time intervals and length measurements are not the same for all observers, but instead depend on their state of motion. In 1908, Minkowski gave a geometric formulation of Einstein’s theory by introducing a pseudo-inner product in the four-dimensional spacetime \(\mathbb {R}^4\). While initially resisting this “excessive mathematization” of his theory, Einstein soon realized that curving spacetime was actually the key to understanding gravity. In 1915, after a long struggle with the mathematics of Riemannian geometry, he was able to arrive at a complete formulation of the general theory of relativity. The predictions of his theory were first confirmed in 1919 by a British solar eclipse expedition, led by Eddington, and have since been verified in every experimental test ever attempted.

Leonor Godinho, José Natário

Chapter 7. Solutions to Selected Exercises

Abstract

In this chapter we present the solutions to 140 selected exercises, chosen among the 333 exercises in the previous chapters.

Leonor Godinho, José Natário

Backmatter

Titel: An Introduction to Riemannian Geometry
verfasst von: Leonor Godinho
José Natário
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-08666-8
Print ISBN: 978-3-319-08665-1
DOI: https://doi.org/10.1007/978-3-319-08666-8