The Decomposition of Global Conformal Invariants (AM-182) / / Spyros Alexakis.
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. Thes...
Saved in:
Superior document: | Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 |
---|---|
VerfasserIn: | |
Place / Publishing House: | Princeton, NJ : : Princeton University Press, , [2012] ©2012 |
Year of Publication: | 2012 |
Edition: | Course Book |
Language: | English |
Series: | Annals of Mathematics Studies ;
182 |
Online Access: | |
Physical Description: | 1 online resource (568 p.) |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
id |
9781400842728 |
---|---|
ctrlnum |
(DE-B1597)447835 (OCoLC)979624183 |
collection |
bib_alma |
record_format |
marc |
spelling |
Alexakis, Spyros, author. aut http://id.loc.gov/vocabulary/relators/aut The Decomposition of Global Conformal Invariants (AM-182) / Spyros Alexakis. Course Book Princeton, NJ : Princeton University Press, [2012] ©2012 1 online resource (568 p.) text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Annals of Mathematics Studies ; 182 Frontmatter -- Contents -- Acknowledgments -- 1. Introduction -- 2. An Iterative Decomposition of Global Conformal Invariants: The First Step -- 3. The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition -- 4. A Result on the Structure of Local Riemannian Invariants: The Fundamental Proposition -- 5. The Inductive Step of the Fundamental Proposition: The Simpler Cases -- 6. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I -- 7. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II -- A. Appendix -- Bibliography -- Index of Authors and Terms -- Index of Symbols restricted access http://purl.org/coar/access_right/c_16ec online access with authorization star 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? Deser and Schwimmer asserted 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. This 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. Key tools used to establish this connection include the Fefferman-Graham ambient metric and the author's super divergence formula. Issued also in print. Mode of access: Internet via World Wide Web. In English. Description based on online resource; title from PDF title page (publisher's Web site, viewed 31. Jan 2022) Conformal invariants. Decomposition (Mathematics). MATHEMATICS / Geometry / Differential. bisacsh CauchyВiemann geometry. DeserГchwimmer conjecture. Khler geometry. Riemannian invariants. Riemannian metrics. Riemannian scalar. Schouten tensor. Weyl tensor. algebraic propositions. ambient metrics. conformal anomalies. conformal invariant. conformal invariants. conformally invariant functionals. curvature tensor. decomposition. differential geometry. global conformal invariant. global invariants. grand conclusion. index theory. induction. iterative decomposition. lemma. lemmas. manifold. theoretical physics. Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 9783110494914 ZDB-23-PMB Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Backlist 2000-2013 9783110442502 print 9780691153476 https://doi.org/10.1515/9781400842728?locatt=mode:legacy https://www.degruyter.com/isbn/9781400842728 Cover https://www.degruyter.com/document/cover/isbn/9781400842728/original |
language |
English |
format |
eBook |
author |
Alexakis, Spyros, Alexakis, Spyros, |
spellingShingle |
Alexakis, Spyros, Alexakis, Spyros, The Decomposition of Global Conformal Invariants (AM-182) / Annals of Mathematics Studies ; Frontmatter -- Contents -- Acknowledgments -- 1. Introduction -- 2. An Iterative Decomposition of Global Conformal Invariants: The First Step -- 3. The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition -- 4. A Result on the Structure of Local Riemannian Invariants: The Fundamental Proposition -- 5. The Inductive Step of the Fundamental Proposition: The Simpler Cases -- 6. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I -- 7. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II -- A. Appendix -- Bibliography -- Index of Authors and Terms -- Index of Symbols |
author_facet |
Alexakis, Spyros, Alexakis, Spyros, |
author_variant |
s a sa s a sa |
author_role |
VerfasserIn VerfasserIn |
author_sort |
Alexakis, Spyros, |
title |
The Decomposition of Global Conformal Invariants (AM-182) / |
title_full |
The Decomposition of Global Conformal Invariants (AM-182) / Spyros Alexakis. |
title_fullStr |
The Decomposition of Global Conformal Invariants (AM-182) / Spyros Alexakis. |
title_full_unstemmed |
The Decomposition of Global Conformal Invariants (AM-182) / Spyros Alexakis. |
title_auth |
The Decomposition of Global Conformal Invariants (AM-182) / |
title_alt |
Frontmatter -- Contents -- Acknowledgments -- 1. Introduction -- 2. An Iterative Decomposition of Global Conformal Invariants: The First Step -- 3. The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition -- 4. A Result on the Structure of Local Riemannian Invariants: The Fundamental Proposition -- 5. The Inductive Step of the Fundamental Proposition: The Simpler Cases -- 6. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I -- 7. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II -- A. Appendix -- Bibliography -- Index of Authors and Terms -- Index of Symbols |
title_new |
The Decomposition of Global Conformal Invariants (AM-182) / |
title_sort |
the decomposition of global conformal invariants (am-182) / |
series |
Annals of Mathematics Studies ; |
series2 |
Annals of Mathematics Studies ; |
publisher |
Princeton University Press, |
publishDate |
2012 |
physical |
1 online resource (568 p.) Issued also in print. |
edition |
Course Book |
contents |
Frontmatter -- Contents -- Acknowledgments -- 1. Introduction -- 2. An Iterative Decomposition of Global Conformal Invariants: The First Step -- 3. The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition -- 4. A Result on the Structure of Local Riemannian Invariants: The Fundamental Proposition -- 5. The Inductive Step of the Fundamental Proposition: The Simpler Cases -- 6. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I -- 7. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II -- A. Appendix -- Bibliography -- Index of Authors and Terms -- Index of Symbols |
isbn |
9781400842728 9783110494914 9783110442502 9780691153476 |
callnumber-first |
Q - Science |
callnumber-subject |
QA - Mathematics |
callnumber-label |
QA646 |
callnumber-sort |
QA 3646 |
url |
https://doi.org/10.1515/9781400842728?locatt=mode:legacy https://www.degruyter.com/isbn/9781400842728 https://www.degruyter.com/document/cover/isbn/9781400842728/original |
illustrated |
Not Illustrated |
dewey-hundreds |
500 - Science |
dewey-tens |
510 - Mathematics |
dewey-ones |
518 - Numerical analysis |
dewey-full |
518 |
dewey-sort |
3518 |
dewey-raw |
518 |
dewey-search |
518 |
doi_str_mv |
10.1515/9781400842728?locatt=mode:legacy |
oclc_num |
979624183 |
work_keys_str_mv |
AT alexakisspyros thedecompositionofglobalconformalinvariantsam182 AT alexakisspyros decompositionofglobalconformalinvariantsam182 |
status_str |
n |
ids_txt_mv |
(DE-B1597)447835 (OCoLC)979624183 |
carrierType_str_mv |
cr |
hierarchy_parent_title |
Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Backlist 2000-2013 |
is_hierarchy_title |
The Decomposition of Global Conformal Invariants (AM-182) / |
container_title |
Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 |
_version_ |
1770176668039643136 |
fullrecord |
<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>05930nam a22010575i 4500</leader><controlfield tag="001">9781400842728</controlfield><controlfield tag="003">DE-B1597</controlfield><controlfield tag="005">20220131112047.0</controlfield><controlfield tag="006">m|||||o||d||||||||</controlfield><controlfield tag="007">cr || ||||||||</controlfield><controlfield tag="008">220131t20122012nju fo d z eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9781400842728</subfield></datafield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1515/9781400842728</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-B1597)447835</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)979624183</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-B1597</subfield><subfield code="b">eng</subfield><subfield code="c">DE-B1597</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">nju</subfield><subfield code="c">US-NJ</subfield></datafield><datafield tag="050" ind1=" " ind2="4"><subfield code="a">QA646</subfield></datafield><datafield tag="072" ind1=" " ind2="7"><subfield code="a">MAT012030</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">518</subfield><subfield code="2">22</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Alexakis, Spyros, </subfield><subfield code="e">author.</subfield><subfield code="4">aut</subfield><subfield code="4">http://id.loc.gov/vocabulary/relators/aut</subfield></datafield><datafield tag="245" ind1="1" ind2="4"><subfield code="a">The Decomposition of Global Conformal Invariants (AM-182) /</subfield><subfield code="c">Spyros Alexakis.</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">Course Book</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Princeton, NJ : </subfield><subfield code="b">Princeton University Press, </subfield><subfield code="c">[2012]</subfield></datafield><datafield tag="264" ind1=" " ind2="4"><subfield code="c">©2012</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">1 online resource (568 p.)</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">computer</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">online resource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="347" ind1=" " ind2=" "><subfield code="a">text file</subfield><subfield code="b">PDF</subfield><subfield code="2">rda</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Annals of Mathematics Studies ;</subfield><subfield code="v">182</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="t">Frontmatter -- </subfield><subfield code="t">Contents -- </subfield><subfield code="t">Acknowledgments -- </subfield><subfield code="t">1. Introduction -- </subfield><subfield code="t">2. An Iterative Decomposition of Global Conformal Invariants: The First Step -- </subfield><subfield code="t">3. The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition -- </subfield><subfield code="t">4. A Result on the Structure of Local Riemannian Invariants: The Fundamental Proposition -- </subfield><subfield code="t">5. The Inductive Step of the Fundamental Proposition: The Simpler Cases -- </subfield><subfield code="t">6. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I -- </subfield><subfield code="t">7. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II -- </subfield><subfield code="t">A. Appendix -- </subfield><subfield code="t">Bibliography -- </subfield><subfield code="t">Index of Authors and Terms -- </subfield><subfield code="t">Index of Symbols</subfield></datafield><datafield tag="506" ind1="0" ind2=" "><subfield code="a">restricted access</subfield><subfield code="u">http://purl.org/coar/access_right/c_16ec</subfield><subfield code="f">online access with authorization</subfield><subfield code="2">star</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">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? Deser and Schwimmer asserted 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. This 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. Key tools used to establish this connection include the Fefferman-Graham ambient metric and the author's super divergence formula.</subfield></datafield><datafield tag="530" ind1=" " ind2=" "><subfield code="a">Issued also in print.</subfield></datafield><datafield tag="538" ind1=" " ind2=" "><subfield code="a">Mode of access: Internet via World Wide Web.</subfield></datafield><datafield tag="546" ind1=" " ind2=" "><subfield code="a">In English.</subfield></datafield><datafield tag="588" ind1="0" ind2=" "><subfield code="a">Description based on online resource; title from PDF title page (publisher's Web site, viewed 31. Jan 2022)</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Conformal invariants.</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Decomposition (Mathematics).</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">MATHEMATICS / Geometry / Differential.</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">CauchyВiemann geometry.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">DeserГchwimmer conjecture.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">Khler geometry.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">Riemannian invariants.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">Riemannian metrics.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">Riemannian scalar.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">Schouten tensor.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">Weyl tensor.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">algebraic propositions.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">ambient metrics.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">conformal anomalies.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">conformal invariant.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">conformal invariants.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">conformally invariant functionals.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">curvature tensor.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">decomposition.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">differential geometry.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">global conformal invariant.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">global invariants.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">grand conclusion.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">index theory.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">induction.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">iterative decomposition.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">lemma.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">lemmas.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">manifold.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">theoretical physics.</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Title is part of eBook package:</subfield><subfield code="d">De Gruyter</subfield><subfield code="t">Princeton Annals of Mathematics eBook-Package 1940-2020</subfield><subfield code="z">9783110494914</subfield><subfield code="o">ZDB-23-PMB</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Title is part of eBook package:</subfield><subfield code="d">De Gruyter</subfield><subfield code="t">Princeton University Press eBook-Package Backlist 2000-2013</subfield><subfield code="z">9783110442502</subfield></datafield><datafield tag="776" ind1="0" ind2=" "><subfield code="c">print</subfield><subfield code="z">9780691153476</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://doi.org/10.1515/9781400842728?locatt=mode:legacy</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://www.degruyter.com/isbn/9781400842728</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="3">Cover</subfield><subfield code="u">https://www.degruyter.com/document/cover/isbn/9781400842728/original</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">978-3-11-044250-2 Princeton University Press eBook-Package Backlist 2000-2013</subfield><subfield code="c">2000</subfield><subfield code="d">2013</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">EBA_BACKALL</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">EBA_CL_MTPY</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">EBA_EBACKALL</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">EBA_EBKALL</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">EBA_ECL_MTPY</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">EBA_EEBKALL</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">EBA_ESTMALL</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">EBA_PPALL</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">EBA_STMALL</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV-deGruyter-alles</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">PDA12STME</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">PDA13ENGE</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">PDA18STMEE</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">PDA5EBK</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-23-PMB</subfield><subfield code="c">1940</subfield><subfield code="d">2020</subfield></datafield></record></collection> |