An Introduction to Riemannian Geometry: With Applications to by José Natário, Leonor Godinho

By José Natário, Leonor Godinho

In contrast to many different texts on differential geometry, this textbook additionally bargains fascinating functions to geometric mechanics and normal relativity.

The first half is a concise and self-contained creation to the fundamentals of manifolds, differential kinds, metrics and curvature. the second one half reviews purposes to mechanics and relativity together with the proofs of the Hawking and Penrose singularity theorems. it may be independently used for one-semester classes in both of those subjects.

The major principles are illustrated and additional built by means of quite a few examples and over three hundred routines. particular suggestions are supplied for lots of of those routines, making An advent to Riemannian Geometry excellent for self-study.

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Extra resources for An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity (Universitext)

Example text

Seven of the ten generators of the group take the same form in both groups: four spatio-temporal translations, expressing (respectively) the homogeneity of the relative space of each inertial frame and the uniformity of the time – absolute or relative – of each inertial frame; and three spatial rotations, expressing the isotropy of the relative space of each inertial frame. They correspond, respectively to the conservation of the linear momentum and energy, and of the angular momentum of the dynamical system.

Both involve a “trace” of the Riemann tensor called the Ricci tensor. But in the case of general relativity, as emphasized above, the affine structure, including its curvature tensor, is completely determined by the metric tensor field. 14. Differentiable Manifolds, Fiber Bundlesll Up to now, I have not discussed the nature of the mathematical space, with which the metric and affine fields of general relativity are associated. In the case of Galilei-Newtonian and special-relativistic space-times, the unique structure of these spaces is determined by the respective kinematical symmetry groups of these theories: The inhomogeneous Galilei group leads to Galilei-Newtonian space-time, while the Poincar´e group leads to Minkowski space-time.

Thirdly, not only is the local structure (local, in the sense of a finite but limited region) of space-time dynamized; the global structure (global in the sense of the entire topology) is no longer given a priori. , on a patch of space-time), one must work out the global topology of the maximally extended manifold(s) – criteria must be given for the selection of such an extension (or extensions if one is not uniquely selected) – compatible with the local space-time structure of that solution. Solutions are possible that are spatially and/or temporally finite but unbounded.

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