Editing Splitting theorem

{{Short description|On when a complete Riemannian manifold with nonnegative Ricci curvature is a product space}}
In the [[mathematics|mathematical]] field of [[differential geometry]], there are various '''splitting theorems''' on when a [[pseudo-Riemannian manifold]] can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of [[Lorentzian manifold]]s.

==Cheeger and Gromoll's Riemannian splitting theorem==
Any connected [[Riemannian manifold]] {{mvar|M}} has an underlying [[metric space]] structure, and this allows the definition of a ''geodesic line'' as a map {{math|''c'': ℝ → ''M''}} such that the distance from {{math|''c''(''s'')}} to {{math|''c''(''t'')}} equals {{math|{{!}} ''t'' − ''s'' {{!}}}} for arbitrary {{mvar|s}} and {{mvar|t}}. This is to say that the restriction of {{mvar|c}} to any bounded interval is a curve of minimal length that connects its endpoints.{{sfnm|1a1=Besse|1y=1987|1loc=Definition 6.64|2a1=Petersen|2y=2016|2p=298|3a1=Schoen|3a2=Yau|3y=1994|3p=12}}

In 1971, [[Jeff Cheeger]] and [[Detlef Gromoll]] proved that if a [[Hopf–Rinow theorem|geodesically complete]] and connected Riemannian manifold of nonnegative [[Ricci curvature]] contains any geodesic line, then it must split isometrically as the product of a complete Riemannian manifold with {{math|ℝ}}. The proof was later simplified by Jost Eschenburg and Ernst Heintze. In 1936, [[Stefan Cohn-Vossen]] had originally formulated and proved the theorem in the case of two-dimensional manifolds, and [[Victor Toponogov]] had extended Cohn-Vossen's work to higher dimensions, under the special condition of nonnegative [[sectional curvature]].{{sfnm|1a1=Besse|1y=1987|1loc=Section 6E|2a1=Petersen|2y=2016|2loc=Theorem 7.3.5|3a1=Schoen|3a2=Yau|3y=1994|3loc=Section 1.2}}

The proof can be summarized as follows.{{sfnm|1a1=Besse|1y=1987|1loc=Section 6G|2a1=Petersen|2y=2016|2loc=Section 7.3|3a1=Schoen|3a2=Yau|3y=1994|3loc=Section 1.2}} The condition of a geodesic line allows for two [[Busemann function]]s to be defined. These can be considered a normalized Riemannian distance function to the two endpoints of the line. From the fundamental ''Laplacian comparison theorem'' proved earlier by [[Eugenio Calabi]], these functions are both [[harmonic function|superharmonic]] under the Ricci curvature assumption. Either of these functions could be negative at some points, but the [[triangle inequality]] implies that their sum is nonnegative. The [[strong maximum principle]] implies that the sum is identically zero and hence that each Busemann function is in fact (weakly) a [[harmonic function]]. [[Weyl's lemma (Laplace equation)|Weyl's lemma]] implies the infinite differentiability of the Busemann functions. Then, the proof can be finished by using [[Bochner's formula]] to construct [[parallel transport|parallel vector field]]s, setting up the [[holonomy group|de Rham decomposition theorem]].{{sfnm|1a1=Schoen|1a2=Yau|1y=1994|1loc=Section 1.2}} Alternatively, the theory of [[Riemannian submersion]]s may be invoked.{{sfnm|1a1=Besse|1y=1987|1p=176}}

As a consequence of their splitting theorem, Cheeger and Gromoll were able to prove that the [[universal cover]] of any [[closed manifold]] of nonnegative Ricci curvature must split isometrically as the product of a closed manifold with a [[Euclidean space]]. If the universal cover is topologically [[contractible]], then it follows that all metrics involved must be [[flat manifold|flat]].{{sfnm|1a1=Petersen|1y=2016|1loc=Section 7.3.3}}

==Lorentzian splitting theorem==
In 1982, [[Shing-Tung Yau]] conjectured that a particular [[Lorentzian manifold|Lorentzian]] version of Cheeger and Gromoll's theorem should hold.{{sfnm|1a1=Yau|1y=1982|1loc=Problem 115}} Proofs in various levels of generality were found by Jost Eschenburg, Gregory Galloway, and Richard Newman. In these results, the role of geodesic completeness is replaced by either the condition of [[globally hyperbolic manifold|global hyperbolicity]] or of ''timelike'' geodesic completeness. The nonnegativity of Ricci curvature is replaced by the ''timelike convergence condition'' that the Ricci curvature is nonnegative in all timelike directions. The geodesic line is required to be timelike.{{sfnm|1a1=Beem|1a2=Ehrlich|1a3=Easley|1y=1996|1loc=Chapter 14}}

==References==
'''Notes.'''
{{Reflist|30em}}
'''Historical articles.'''
{{refbegin}}
*{{cite journal |first1=Jeff |last1=Cheeger |author-link1=Jeff Cheeger|author-link2=Detlef Gromoll|first2=Detlef |last2=Gromoll |title=The splitting theorem for manifolds of nonnegative Ricci curvature |journal=[[Journal of Differential Geometry]] |volume=6 |year=1971 |issue=1 |pages=119–128 |mr=0303460 |doi=10.4310/jdg/1214430220 |doi-access=free|zbl=0223.53033}}
*{{cite journal |first=S. |last=Cohn-Vossen |title=Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken |journal=[[Matematicheskii Sbornik]] |volume=43 |issue=2 |year=1936 |pages=139–163|url=https://eudml.org/doc/64812|author-link1=Stefan Cohn-Vossen|zbl=0014.27601|jfm=62.0862.01}}
<!--*{{cite journal |last=Eschenburg |first=J.-H. |title=The splitting theorem for space-times with strong energy condition|zbl=0647.53043 |journal=[[Journal of Differential Geometry]] |volume=27 |year=1988 |issue=3 |pages=477–491 |doi=10.4310/jdg/1214442005 |doi-access=free|mr=0940115 }}
*{{cite journal |last=Galloway |first=Gregory J. |title=The Lorentzian splitting theorem without the completeness assumption|zbl=0667.53048 |journal=[[Journal of Differential Geometry]]|mr=0982181 |volume=29 |year=1989 |issue=2 |pages=373–387 |doi=10.4310/jdg/1214442881 |doi-access=free }}
*{{cite journal |last=Newman |first=Richard P. A. C. |mr=1030669|title=A proof of the splitting conjecture of S.-T. Yau |journal=[[Journal of Differential Geometry]] |volume=31 |year=1990 |issue=1 |pages=163–184 |doi=10.4310/jdg/1214444093 |doi-access=free|zbl=0695.53049 }}-->
*{{cite journal |last=Toponogov |first=V. A. |author-link1=Victor Toponogov|title=Riemannian spaces which contain straight lines |zbl=0138.42902|journal=American Mathematical Society Translations|series=Second Series|volume=37|year=1964 |pages=287–290|doi=10.1090/trans2/037|issue=Twenty-two papers on algebra, number theory and differential geometry|isbn=978-0-8218-1737-7 |translator-last1=Robinson|translator-first1=A.}}
*{{cite journal|doi=10.1090/trans2/070|last=Toponogov |first=V. A. |author-link1=Victor Toponogov|title=The metric structure of Riemannian spaces with nonnegative curvature which contain straight lines|year=1968|journal=American Mathematical Society Translations|series=Second Series|volume=70|issue=Thirty-one invited addresses (eight in abstract) at the International Congress of Mathematicians in Moscow, 1966|translator-last1=West|translator-first1=A.|pages=225–239|isbn=978-0-8218-1770-4 |zbl=0187.43801}}
*{{wikicite|ref={{sfnRef|Yau|1982}}|reference={{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=978-1-4008-8191-8 |mr=0645762|author-link1=Shing-Tung Yau|editor-link1=Shing-Tung Yau|zbl=0479.53001|ref=none}} Reprinted in {{harvtxt|Schoen|Yau|1994}}.}}
{{refend}}
'''Textbooks.'''
{{refbegin}}
*{{cite book|last1=Beem|first1=John K.|last2=Ehrlich|first2=Paul E.|last3=Easley|first3=Kevin L.|title=Global Lorentzian geometry|doi=10.1201/9780203753125|edition=Second edition of 1981 original|series=Monographs and Textbooks in Pure and Applied Mathematics|volume=202|publisher=[[Marcel Dekker, Inc.]]|location=New York|year=1996|isbn=0-8247-9324-2|mr=1384756|zbl=0846.53001}}
* {{cite book|last1=Besse|first1=Arthur L.|title=Einstein manifolds|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3)|volume=10|publisher=[[Springer-Verlag]]|location=Berlin|year=1987|isbn=3-540-15279-2|mr=0867684|others=Reprinted in 2008|doi=10.1007/978-3-540-74311-8|author-link1=Arthur Besse|zbl=0613.53001}}
*{{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|last1=Schoen|first1=R.|author-link1=Richard Schoen|author-link2=Shing-Tung Yau|last2=Yau|first2=S.-T.|title=Lectures on differential geometry|translator-last1=Ding|translator-first1=Wei Yue|translator-last2=Cheng|translator-first2=S. Y.|translator-link2=Shiu-Yuen Cheng|series=Conference Proceedings and Lecture Notes in Geometry and Topology|volume=1|publisher=International Press|location=Cambridge, MA|year=1994|isbn=1-57146-012-8|mr=1333601|zbl=0830.53001}}
{{refend}}

[[Category:Theorems in Riemannian geometry]]