Read e-book online An Introduction to Riemannian Geometry: With Applications to PDF

By José Natário, Leonor Godinho

ISBN-10: 3319086669

ISBN-13: 9783319086668

Not like many different texts on differential geometry, this textbook additionally deals attention-grabbing purposes to geometric mechanics and basic 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 experiences functions 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 rules are illustrated and extra constructed by way of quite a few examples and over three hundred routines. distinct ideas are supplied for plenty of of those routines, making An creation to Riemannian Geometry perfect for self-study.

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Additional info for An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity (Universitext)

Example text

Hint : Show that if a is a constant satisfying −1 < a < 1, 1 + β1 β 2 x+a is increasing on −1 ≤ x ≤ 1. then the function f (x) = 1 + ax β1 + β 2 (b) Λ(β1 )Λ(β2 ) = Λ . 14 that the composition of two boosts in the x1 -direction is another boost in the x1 -direction. Since Λ−1 (β) = Λ(−β) the collection of all such special Lorentz transformations forms a subgroup of L. We point out, however, that the composition of two boosts in two different directions is, in general, not equivalent to a single boost in any direction.

3m. However, it is found experimentally that the beam reaches one-half intensity after travelling approximately 38m. Explain! 29). Draw a Minkowski diagram which represents the situation. 12 and, in particular, to the line x ˆ4 = 1. Each point on this line can be identified with the appearance of the reading “1” on a clock that is stationary at some point in ˆ . These all occur “simulˆ because his clocks have been synchronized. However, each taneously” for O of these events occurs at a different “time” in S so O will disagree.

As it is with time dilation, the correct physical interpretation of the Lorentz contraction often requires rather subtle and delicate argument. 26 Imagine a barn which, at rest, measures 8 meters in length. A (very fast) runner carries a pole of rest length 16 meters toward the barn at such a high speed that, for an observer at rest with the barn, it appears Lorentz contracted to 8 meters and therefore fits inside the barn. This observer slams the front door shut at the instant the back of the pole enters the front of the barn and so encloses the pole entirely within the barn.

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An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity (Universitext) by José Natário, Leonor Godinho

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