Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question ...
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This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? This book asserts that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern–Gauss–Bonnet integrand. The book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants—such as the classical Riemannian invariants and the more recently studied conformal invariants—and the study of global invariants, in this case conformally invariant integrals.Less
This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? This book asserts that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern–Gauss–Bonnet integrand. The book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants—such as the classical Riemannian invariants and the more recently studied conformal invariants—and the study of global invariants, in this case conformally invariant integrals.
Charles Fefferman and C. Robin Graham
- 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 ...
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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 conformal structure on a manifold is an equivalence class of Riemannian metrics, in which two metrics are identified if one is a positive smooth multiple of the other. In [FG], the authors outlined a construction of a nondegenerate Lorentz metric in n+2 dimensions associated to an n-dimensional conformal manifold, which they called the ambient metric. This association enables one to construct conformal invariants in n dimensions from pseudo-Riemannian invariants in n+2 dimensions, and in particular shows that conformal invariants are plentiful. The formal theory outlined in [FG] did not provide details. This book provides these details. An overview of the subsequent chapters is also presented.Less
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 conformal structure on a manifold is an equivalence class of Riemannian metrics, in which two metrics are identified if one is a positive smooth multiple of the other. In [FG], the authors outlined a construction of a nondegenerate Lorentz metric in n+2 dimensions associated to an n-dimensional conformal manifold, which they called the ambient metric. This association enables one to construct conformal invariants in n dimensions from pseudo-Riemannian invariants in n+2 dimensions, and in particular shows that conformal invariants are plentiful. The formal theory outlined in [FG] did not provide details. This book provides these details. An overview of the subsequent chapters is also presented.
Paula Tretkoff
- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0003
- Subject:
- Mathematics, Geometry / Topology
This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and ...
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This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.Less
This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.
Charles Fefferman and C. Robin Graham
- 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 ...
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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 relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on tensors. It is assumed throughout this chapter that n ≥ 3.Less
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 relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on tensors. It is assumed throughout this chapter that n ≥ 3.
Loring W. Tu
- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0013
- Subject:
- Mathematics, Educational Mathematics
This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a ...
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This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω on M, by averaging all the left translates of ω over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.Less
This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω on M, by averaging all the left translates of ω over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.
