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Pseudo-Riemannian Homogeneous Structures

Part of the Developments in Mathematics Book Series (volume 59)

DOI: 10.1007/978-3-030-18152-9

ISBN: 978-3-030-18151-2

Publisher: Springer International Publishing - Springer Journals

Aug 15, 2019

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Chapter 1G-structures, Holonomy and Homogeneous Spaces

Calvaruso, Giovanni; Castrillón López, Marco

Pages: 1 –39

[In the following sections we establish the foundations needed for the subsequent chapters. The reader is assumed to be familiar with the notions of smooth manifolds, tensorial algebra, pseudo-Riemannian geometry, Lie groups and actions of groups. From this starting point, we introduce principal...
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Chapter 2Ambrose–Singer Connections and Homogeneous Spaces

Calvaruso, Giovanni; Castrillón López, Marco

Pages: 41 –58

[Homogeneous and locally homogeneous spaces are among the most important objects of study in Differential Geometry. They have been extensively investigated using several methods and techniques. When considering a homogeneous space, many geometric properties translate into algebraic properties....
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Chapter 3Locally Homogeneous Pseudo-Riemannian Manifolds

Calvaruso, Giovanni; Castrillón López, Marco

Pages: 59 –90

[The conditions on a pseudo-Riemannian manifold (M, g) appearing in the statement of the Ambrose–Singer Theorem (Theorem 2.2.1 in Chap. 2) can be classified into two different types.]
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Chapter 4Classification of Homogeneous Structures

Calvaruso, Giovanni; Castrillón López, Marco

Pages: 91 –131

[In previous chapters we have seen how AS-connections and ASK-connections can be used to study homogeneous and locally homogeneous spaces.]
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Chapter 5Homogeneous Structures of Linear Type

Calvaruso, Giovanni; Castrillón López, Marco

Pages: 133 –169

[Among the different applications and possibilities of homogeneous structures, the translation of the algebraic classifications of the tensors (like those provided in the previous chapter) into geometric terms might be the most important one. For example, as we mentioned at the end of the...
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Chapter 6Reduction of Homogeneous Structures

Calvaruso, Giovanni; Castrillón López, Marco

Pages: 171 –195

[Symmetries represent a classical tool in reduction procedures, found in many different contexts, such as systems of differential equations, variational principles, symplectic or other geometric structures, etc. The philosophy is quite simple: given any object on a manifold...
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Chapter 7Where All This Fails: Non-reductive Homogeneous Pseudo-Riemannian Manifolds

Calvaruso, Giovanni; Castrillón López, Marco

Pages: 197 –221

[As shown in the previous chapters, reductive homogeneous pseudo-Riemannian manifolds are completely characterized by the existence of a homogeneous pseudo-Riemannian structure, and the geometric properties of these spaces are reflected by the properties of such a structure.]

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