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## Sasakian Geometry

*Charles Boyer and Krzysztof Galicki*

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.001.0001
- Subject:
- Mathematics, Geometry / Topology

Sasakian manifolds were first introduced in 1962. This book's main focus is on the intricate relationship between Sasakian and Kähler geometries, especially when the Kähler structure is that of an ... More

## Foliations

*Charles P. Boyer and Krzysztof Galicki*

### in Sasakian Geometry

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0003
- Subject:
- Mathematics, Geometry / Topology

As Sasakian manifolds are all examples of Riemannian foliations with one-dimensional leaves, this chapter goes into the world of foliations with a particular focus on the Riemannian case. Haeiger ... More

## Harmonic Morphisms Between Riemannian Manifolds

*Paul Baird and John C. Wood*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.001.0001
- Subject:
- Mathematics, Pure Mathematics

Harmonic morphisms are maps which preserve Laplace's equation. More explicitly, a map between Riemannian manifolds is called a harmonic morphism if its composition with any locally defined harmonic ... More

## Riemannian manifolds and conformality

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0002
- Subject:
- Mathematics, Pure Mathematics

This chapter introduces fundamental concepts from Riemannian geometry, including the Laplacian and harmonic functions on a Riemannian manifold. The next two sections discuss weak conformality and the ... More

## Fundamental properties of harmonic morphisms

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0004
- Subject:
- Mathematics, Pure Mathematics

This chapter characterizes harmonic morphisms between Riemannian manifolds and discusses their basic properties. The first non-constant term of the Taylor series of a harmonic morphism at a point ... More

## Preliminaries

*Sergiu Klainerman and Jérémie Szeftel*

### in Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations: (AMS-210)

- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691212425
- eISBN:
- 9780691218526
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691212425.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses the main quantities, equations, and basic tools needed in the following chapters. It is the main reference kit providing all main null structure and null Bianchi equations, in ... More

## GCM Procedure

*Sergiu Klainerman and Jérémie Szeftel*

### in Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations: (AMS-210)

- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691212425
- eISBN:
- 9780691218526
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691212425.003.0009
- Subject:
- Mathematics, Geometry / Topology

This chapter describes the general covariant modulation (GCM) procedure in detail. It considers an axially symmetric polarized spacetime region R foliated by two functions (u, s) such that: on R, (u, ... More

## Entropy for hyperbolic Riemann surface laminations II

*Tien-Cuong Dinh, Viet-Anh Nguyen, and Nessim Sibony*

*Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)*

### in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0021
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated ... More

## Thurston's Proof

*Benson Farb and Dan Margalit*

### in A Primer on Mapping Class Groups (PMS-49)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691147949
- eISBN:
- 9781400839049
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147949.003.0016
- Subject:
- Mathematics, Geometry / Topology

This chapter describes Thurston's original path of discovery to the Nielsen–Thurston classification theorem. It first provides an example that illustrates much of the general theory, focusing on ... More

## Flat connections and holonomy

*Clifford Henry Taubes*

### in Differential Geometry: Bundles, Connections, Metrics and Curvature

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0013
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines flat connections. A connection on a principal bundle is said to be flat when its curvature 2-form is identically zero. The discussions cover flat connections on bundles over the ... More

## Entropy for hyperbolic Riemann surface laminations I

*Tien-Cuong Dinh, Viet-Anh Nguyen, and Nessim Sibony*

*Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)*

### in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0020
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of ... More

## Teichmuller Geometry

*Benson Farb and Dan Margalit*

### in A Primer on Mapping Class Groups (PMS-49)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691147949
- eISBN:
- 9781400839049
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147949.003.0012
- Subject:
- Mathematics, Geometry / Topology

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines ... More

## Pseudo-Anosov Theory

*Benson Farb and Dan Margalit*

### in A Primer on Mapping Class Groups (PMS-49)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691147949
- eISBN:
- 9781400839049
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147949.003.0015
- Subject:
- Mathematics, Geometry / Topology

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping ... More

## Magmatic fabrics, structures and microstructures

*Jean-Luc Bouchez and Adolphe Nicolas*

### in Principles of Rock Deformation and Tectonics

- Published in print:
- 2021
- Published Online:
- October 2021
- ISBN:
- 9780192843876
- eISBN:
- 9780191926501
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192843876.003.0007
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

A magma is a two-phase material made of crystals immersed in a silicate melt, which displays a high viscosity contrast between the liquid and the solid fractions. A specific rheological behaviour is ... More

## Autonomous Geometrical Mechanics

*Peter Mann*

### in Lagrangian and Hamiltonian Dynamics

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0022
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic ... More

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