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## Introduction

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0001
- Subject:
- Mathematics, Geometry / Topology

This introductory chapter begins with a brief definition of conformal geometry. Conformal geometry is the study of spaces in which one knows how to measure infinitesimal angles but not lengths. A ... More

## The Ambient Metric (AM-178)

*Charles Fefferman and C. Robin Graham*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient ... More

## Jet Isomorphism

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0008
- Subject:
- Mathematics, Geometry / Topology

A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered ... More

## Scalar Invariants

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0009
- Subject:
- Mathematics, Geometry / Topology

This chapter shows how to derive a characterization of scalar invariants of conformal structures by reduction to the relevant results of [BEGr]. In [FG], the authors conjectured that when n is odd, ... More

## Conformally Flat and Conformally Einstein Spaces

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0007
- Subject:
- Mathematics, Geometry / Topology

This chapter analyzes the ambient and Poincaré metrics for locally conformally flat manifolds and for conformal classes containing an Einstein metric. The obstruction tensor vanishes for even ... More

## Poincaré Metrics

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0004
- Subject:
- Mathematics, Geometry / Topology

This chapter considers the formal theory for Poincaré metrics associated to a conformal manifold (M, [g]). It shows that even Poincaré metrics are in one-to-one correspondence with straight ambient ... More

## Self-dual Poincaré Metrics

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0005
- Subject:
- Mathematics, Geometry / Topology

As an application of the formal theory for Poincaré metrics, this chapter presents a formal power series proof of a result of LeBrun [LeB] asserting the existence and uniqueness of a real-analytic ... More

## Conformai Curvature Tensors

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0006
- Subject:
- Mathematics, Geometry / Topology

This chapter studies conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form ... More

## Shape Dynamics: Relativity and Relationalism

*Flavio Mercati*

- Published in print:
- 2018
- Published Online:
- April 2018
- ISBN:
- 9780198789475
- eISBN:
- 9780191831294
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198789475.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Particle Physics / Astrophysics / Cosmology

Shape Dynamics is a new theory of gravity that is based on fewer and more fundamental first principles than General Relativity. The most important feature of this theory is the replacement of ... More

## Ambient Metrics

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter presents the full infinite-order formal theory for ambient metric forms, including the freedom at order n/2 in all dimensions and the precise description of the log terms when n ≤ 4 is ... More

## Formal Theory

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics ... More

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