Soul theorem: Difference between revisions
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==Soul theorem== | ==Soul theorem== | ||
Cheeger and Gromoll's '''soul theorem''' states:{{sfnm|1a1=Cheeger|1a2=Ebin|1y=2008|1loc=Chapter 8|2a1=Petersen|2y=2016|2loc=Theorem 12.4.1|3a1=Sakai|3y=1996|3loc=Theorem V.3.4}} | Cheeger and Gromoll's '''soul theorem''' states:{{sfnm|1a1=Cheeger|1a2=Ebin|1y=2008|1loc=Chapter 8|2a1=Petersen|2y=2016|2loc=Theorem 12.4.1|3a1=Sakai|3y=1996|3loc=Theorem V.3.4}} | ||
:If {{math|(''M'', ''g'')}} is a [[complete space|complete]] [[connected space|connected]] [[Riemannian manifold]] with nonnegative [[sectional curvature]], then there exists a [[closed manifold|closed]] [[Glossary of Riemannian and metric geometry#T|totally convex]], [[Glossary of Riemannian and metric geometry#T|totally geodesic]] [[embedded submanifold]] whose [[Glossary of Riemannian and metric geometry#N|normal bundle]] is [[diffeomorphism|diffeomorphic]] to {{math|''M''}}. | :If {{math|(''M'', ''g'')}} is a [[complete space|complete]] [[connected space|connected]] and non compact [[Riemannian manifold]] with nonnegative [[sectional curvature]], then there exists a [[closed manifold|closed]] [[Glossary of Riemannian and metric geometry#T|totally convex]], [[Glossary of Riemannian and metric geometry#T|totally geodesic]] [[embedded submanifold]] whose [[Glossary of Riemannian and metric geometry#N|normal bundle]] is [[diffeomorphism|diffeomorphic]] to {{math|''M''}}. | ||
Such a submanifold is called a '''soul''' of {{math|(''M'', ''g'')}}. By the [[Gauss-Codazzi equations|Gauss equation]] and total geodesicity, the induced Riemannian metric on the soul automatically has nonnegative sectional curvature. Gromoll and Meyer had earlier studied the case of positive sectional curvature, where they showed that a soul is given by a single point, and hence that {{mvar|M}} is diffeomorphic to [[Euclidean space]].{{sfnm|1a1=Petersen|1y=2016|1p=462|2a1=Sakai|2y=1996|2loc=Corollary V.3.5}} | Such a submanifold is called a '''soul''' of {{math|(''M'', ''g'')}}. By the [[Gauss-Codazzi equations|Gauss equation]] and total geodesicity, the induced Riemannian metric on the soul automatically has nonnegative sectional curvature. Gromoll and Meyer had earlier studied the case of positive sectional curvature, where they showed that a soul is given by a single point, and hence that {{mvar|M}} is diffeomorphic to [[Euclidean space]].{{sfnm|1a1=Petersen|1y=2016|1p=462|2a1=Sakai|2y=1996|2loc=Corollary V.3.5}} | ||
Very simple examples, as below, show that the soul is not uniquely determined by {{math|(''M'', ''g'')}} in general. However, Vladimir Sharafutdinov constructed a [[Lipschitz map|1-Lipschitz]] [[Retraction (topology)|retraction]] from {{mvar|M}} to any of its souls, thereby showing that any two souls are [[Isometry|isometric]]. This mapping is known as | Very simple examples, as below, show that the soul is not uniquely determined by {{math|(''M'', ''g'')}} in general. However, Vladimir Sharafutdinov constructed a [[Lipschitz map|1-Lipschitz]] [[Retraction (topology)|retraction]] from {{mvar|M}} to any of its souls, thereby showing that any two souls are [[Isometry|isometric]]. This mapping is known as [[Sharafutdinov's retraction]].{{sfnm|1a1=Chow |1a2=Chu|1a3=Glickenstein|1a4=Guenther|1y=2010|1loc=Theorem I.25}} | ||
Cheeger and Gromoll also posed the converse question of whether there is a complete Riemannian metric of nonnegative sectional curvature on the total space of any [[vector bundle]] over a closed manifold of positive sectional curvature.{{sfnm|1a1=Yau|1y=1982|1loc=Problem 6}} The answer is now known to be negative, although the existence theory is not fully understood.{{sfnm|1a1=Petersen|1y=2016|1p=469}} | Cheeger and Gromoll also posed the converse question of whether there is a complete Riemannian metric of nonnegative sectional curvature on the total space of any [[vector bundle]] over a closed manifold of positive sectional curvature.{{sfnm|1a1=Yau|1y=1982|1loc=Problem 6}} The answer is now known to be negative, although the existence theory is not fully understood.{{sfnm|1a1=Petersen|1y=2016|1p=469}} | ||
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'''Sources.''' | '''Sources.''' | ||
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*{{cite book|mr=2394158|last1=Cheeger|first1=Jeff|last2=Ebin|first2=David G.|title=Comparison theorems in Riemannian geometry|edition=Revised reprint of the 1975 original|publisher=[[Chelsea Publishing Company|AMS Chelsea Publishing]]|location=Providence, RI|year=2008|isbn=978-0-8218-4417-5|author-link1=Jeff Cheeger|author-link2=David Gregory Ebin|doi=10.1090/chel/365}} | *{{cite book |mr=2394158 |last1=Cheeger |first1=Jeff |last2=Ebin |first2=David G. |title=Comparison theorems in Riemannian geometry |edition=Revised reprint of the 1975 original |publisher=[[Chelsea Publishing Company|AMS Chelsea Publishing]] |location=Providence, RI |year=2008 |isbn=978-0-8218-4417-5 |author-link1=Jeff Cheeger |author-link2=David Gregory Ebin |doi=10.1090/chel/365}} | ||
*{{Cite journal | last1=Cheeger | first1=Jeff |author-link1=Jeff Cheeger|author-link2=Detlef Gromoll| last2=Gromoll | first2=Detlef | title=On the structure of complete manifolds of nonnegative curvature |doi=10.2307/1970819 |mr=0309010 | year=1972 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=96 | issue=3 | pages=413–443| jstor=1970819 | url=http://projecteuclid.org/euclid.bams/1183530120 }} | *{{Cite journal |last1=Cheeger |first1=Jeff |author-link1=Jeff Cheeger |author-link2=Detlef Gromoll |last2=Gromoll |first2=Detlef |title=On the structure of complete manifolds of nonnegative curvature |doi=10.2307/1970819 |mr=0309010 |year=1972 |journal=[[Annals of Mathematics]] |series=Second Series |issn=0003-486X |volume=96 |issue=3 |pages=413–443 |jstor=1970819 |url=http://projecteuclid.org/euclid.bams/1183530120}} | ||
*{{cite book|last1=Chow|first1=Bennett|last2=Chu|first2=Sun-Chin|last3=Glickenstein|first3=David|last4=Guenther|first4=Christine|last5=Isenberg|first5=James|last6=Ivey|first6=Tom|last7=Knopf|first7=Dan|last8=Lu|first8=Peng|last9=Luo|first9=Feng|last10=Ni|first10=Lei|title=The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects|series=[[Mathematical Surveys and Monographs]]|volume=163|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2010|doi=10.1090/surv/163|author-link4=Christine Guenther|author-link5=James Isenberg|mr=2604955|isbn=978-0-8218-4661-2}} | *{{cite book |last1=Chow |first1=Bennett |last2=Chu |first2=Sun-Chin |last3=Glickenstein |first3=David |last4=Guenther |first4=Christine |last5=Isenberg |first5=James |last6=Ivey |first6=Tom |last7=Knopf |first7=Dan |last8=Lu |first8=Peng |last9=Luo |first9=Feng |last10=Ni |first10=Lei |title=The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects |series=[[Mathematical Surveys and Monographs]] |volume=163 |publisher=[[American Mathematical Society]] |location=Providence, RI |year=2010 |doi=10.1090/surv/163 |author-link4=Christine Guenther |author-link5=James Isenberg |mr=2604955 |isbn=978-0-8218-4661-2 |author-link10=Lei Ni}} | ||
*{{Cite journal | last1=Gromoll | first1=Detlef | author-link1=Detlef Gromoll|last2=Meyer | first2=Wolfgang | title=On complete open manifolds of positive curvature |doi=10.2307/1970682 |mr=0247590 | year=1969 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=90 | issue=1 | pages=75–90| jstor=1970682 | s2cid=122543838 }} | *{{Cite journal |last1=Gromoll |first1=Detlef |author-link1=Detlef Gromoll |last2=Meyer |first2=Wolfgang |title=On complete open manifolds of positive curvature |doi=10.2307/1970682 |mr=0247590 |year=1969 |journal=[[Annals of Mathematics]] |series=Second Series |issn=0003-486X |volume=90 |issue=1 |pages=75–90 |jstor=1970682 |s2cid=122543838}} | ||
*{{Cite journal| last1=Perelman | first1=Grigori | author1-link=Grigori Perelman | title=Proof of the soul conjecture of Cheeger and Gromoll | url=https://projecteuclid.org/download/pdf_1/euclid.jdg/1214455292 | mr=1285534 | zbl = 0818.53056 | year=1994 | journal=[[Journal of Differential Geometry]] | issn=0022-040X | volume=40 | issue=1 | pages=209–212| doi=10.4310/jdg/1214455292 | doi-access=free }} | *{{Cite journal |last1=Perelman |first1=Grigori |author1-link=Grigori Perelman |title=Proof of the soul conjecture of Cheeger and Gromoll |url=https://projecteuclid.org/download/pdf_1/euclid.jdg/1214455292 |mr=1285534 |zbl=0818.53056 |year=1994 |journal=[[Journal of Differential Geometry]] |issn=0022-040X |volume=40 |issue=1 |pages=209–212 |doi=10.4310/jdg/1214455292 |doi-access=free}} | ||
*{{cite book|last1=Petersen|first1=Peter|title=Riemannian geometry|edition=Third edition of 1998 original|series=[[Graduate Texts in Mathematics]]|volume=171|publisher=[[Springer Publishing|Springer, Cham]]|year=2016|isbn=978-3-319-26652-7|mr=3469435|doi=10.1007/978-3-319-26654-1|zbl=1417.53001}} | *{{cite book |last1=Petersen |first1=Peter |title=Riemannian geometry |edition=Third edition of 1998 original |series=[[Graduate Texts in Mathematics]] |volume=171 |publisher=[[Springer Publishing|Springer, Cham]] |year=2016 |isbn=978-3-319-26652-7 |mr=3469435 |doi=10.1007/978-3-319-26654-1 |zbl=1417.53001}} | ||
*{{cite book|mr=1390760|last1=Sakai|first1=Takashi|title=Riemannian geometry|series=Translations of Mathematical Monographs|volume=149|publisher=[[American Mathematical Society]]|location=Providence, RI|year=1996|isbn=0-8218-0284-4|doi=10.1090/mmono/149|zbl=0886.53002}} | *{{cite book |mr=1390760 |last1=Sakai |first1=Takashi |title=Riemannian geometry |series=Translations of Mathematical Monographs |volume=149 |publisher=[[American Mathematical Society]] |location=Providence, RI |year=1996 |isbn=0-8218-0284-4 |doi=10.1090/mmono/149 |zbl=0886.53002}} | ||
*{{Cite journal|first=V. A.|last=Sharafutdinov|title=Convex sets in a manifold of nonnegative curvature|journal=Mathematical Notes|volume=26|number=1|year=1979 |pages=556–560 |doi=10.1007/BF01140282|s2cid=119764156 }} | *{{Cite journal |first=V. A. |last=Sharafutdinov |title=Convex sets in a manifold of nonnegative curvature |journal=Mathematical Notes |volume=26 |number=1 |year=1979 |pages=556–560 |doi=10.1007/BF01140282 |s2cid=119764156}} | ||
*{{cite encyclopedia|last1=Yau|first1=Shing Tung|title=Problem section|encyclopedia=Seminar on Differential Geometry|pages=669–706|series=Annals of Mathematics Studies|volume=102|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1982|editor-last1=Yau|editor-first1=Shing-Tung|doi=10.1515/9781400881918-035|isbn=9781400881918 |mr=0645762|author-link1=Shing-Tung Yau|editor-link1=Shing-Tung Yau|zbl=0479.53001}} | *{{cite encyclopedia |last1=Yau |first1=Shing Tung |title=Problem section |encyclopedia=Seminar on Differential Geometry |pages=669–706 |series=Annals of Mathematics Studies |volume=102 |publisher=[[Princeton University Press]] |location=Princeton, NJ |year=1982 |editor-last1=Yau |editor-first1=Shing-Tung |doi=10.1515/9781400881918-035 |isbn=9781400881918 |mr=0645762 |author-link1=Shing-Tung Yau |editor-link1=Shing-Tung Yau |zbl=0479.53001}} | ||
{{Refend}} | {{Refend}} | ||
Latest revision as of 17:36, 28 December 2025
Template:Short description In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related soul conjecture, formulated by Cheeger and Gromoll at that time, was proved twenty years later by Grigori Perelman.
Soul theorem
Cheeger and Gromoll's soul theorem states:Template:Sfnm
- If (M, g)Script error: No such module "Check for unknown parameters". is a complete connected and non compact Riemannian manifold with nonnegative sectional curvature, then there exists a closed totally convex, totally geodesic embedded submanifold whose normal bundle is diffeomorphic to MScript error: No such module "Check for unknown parameters"..
Such a submanifold is called a soul of (M, g)Script error: No such module "Check for unknown parameters".. By the Gauss equation and total geodesicity, the induced Riemannian metric on the soul automatically has nonnegative sectional curvature. Gromoll and Meyer had earlier studied the case of positive sectional curvature, where they showed that a soul is given by a single point, and hence that Template:Mvar is diffeomorphic to Euclidean space.Template:Sfnm
Very simple examples, as below, show that the soul is not uniquely determined by (M, g)Script error: No such module "Check for unknown parameters". in general. However, Vladimir Sharafutdinov constructed a 1-Lipschitz retraction from Template:Mvar to any of its souls, thereby showing that any two souls are isometric. This mapping is known as Sharafutdinov's retraction.Template:Sfnm
Cheeger and Gromoll also posed the converse question of whether there is a complete Riemannian metric of nonnegative sectional curvature on the total space of any vector bundle over a closed manifold of positive sectional curvature.Template:Sfnm The answer is now known to be negative, although the existence theory is not fully understood.Template:Sfnm
Examples.
- As can be directly seen from the definition, every compact manifold is its own soul. For this reason, the theorem is often stated only for non-compact manifolds.
- As a very simple example, take MScript error: No such module "Check for unknown parameters". to be Euclidean space RnScript error: No such module "Check for unknown parameters".. The sectional curvature is 0Script error: No such module "Check for unknown parameters". everywhere, and any point of MScript error: No such module "Check for unknown parameters". can serve as a soul of MScript error: No such module "Check for unknown parameters"..
- Now take the paraboloid M = {(x, y, z) : z = x2 + y2Script error: No such module "Check for unknown parameters".}, with the metric gScript error: No such module "Check for unknown parameters". being the ordinary Euclidean distance coming from the embedding of the paraboloid in Euclidean space R3Script error: No such module "Check for unknown parameters".. Here the sectional curvature is positive everywhere, though not constant. The origin (0, 0, 0)Script error: No such module "Check for unknown parameters". is a soul of MScript error: No such module "Check for unknown parameters".. Not every point xScript error: No such module "Check for unknown parameters". of MScript error: No such module "Check for unknown parameters". is a soul of MScript error: No such module "Check for unknown parameters"., since there may be geodesic loops based at xScript error: No such module "Check for unknown parameters"., in which case wouldn't be totally convex.Template:Sfnm
- One can also consider an infinite cylinder M = {(x, y, z) : x2 + y2 = 1Script error: No such module "Check for unknown parameters".}, again with the induced Euclidean metric. The sectional curvature is 0Script error: No such module "Check for unknown parameters". everywhere. Any "horizontal" circle {(x, y, z) : x2 + y2 = 1Script error: No such module "Check for unknown parameters".} with fixed zScript error: No such module "Check for unknown parameters". is a soul of MScript error: No such module "Check for unknown parameters".. Non-horizontal cross sections of the cylinder are not souls since they are neither totally convex nor totally geodesic.Template:Sfnm
Soul conjecture
As mentioned above, Gromoll and Meyer proved that if Template:Mvar has positive sectional curvature then the soul is a point. Cheeger and Gromoll conjectured that this would hold even if Template:Mvar had nonnegative sectional curvature, with positivity only required of all sectional curvatures at a single point.Template:Sfnm This soul conjecture was proved by Grigori Perelman, who established the more powerful fact that Sharafutdinov's retraction is a Riemannian submersion, and even a submetry.Template:Sfnm
References
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