Table Of ContentClassic Texts in the Sciences
Jürgen Jost
Editor
Bernhard Riemann
On the Hypotheses
Which Lie at the
Bases of Geometry
Bernhard Riemann
On the Hypotheses Which Lie at the Bases
of Geometry
Classic Texts in the Sciences
SeriesEditors
OlafBreidbach((cid:2))
JürgenJost
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Jürgen Jost
Editor
Bernhard Riemann
On the Hypotheses Which
Lie at the Bases of Geometry
Editor
JuRrgenJost
MaxPlanckInstituteforMathematicsin
theSciences
Leipzig,Germany
ISSN2365-9963 ISSN2365-9971 (electronic)
ClassicTextsintheSciences
ISBN978-3-319-26040-2 ISBN978-3-319-26042-6 (eBook)
DOI10.1007/978-3-319-26042-6
LibraryofCongressControlNumber:2016936876
MathematicsSubjectClassification(2010):51-03,53-03,53B20,53Z05,00A30,01A55,83-03
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Preface
Thiscouldbetheplotofanovel:Themaincharacterisashy,sickly,youngmathematician,
livinginpoorconditionsataGermanuniversityinthemiddleofthenineteenthcentury.He
doesnotsucceedinestablishingaclosercontactwiththegreatestmathematicalluminary
of his time. He is working toward his habilitation degree (a prerequisite for becoming
a candidate for a professorship at a university in Germany). Part of the process is a
colloquium.For such a habilitation colloquium,the candidatemust proposethree topics
from which the faculty can choose. The first two topics are derived from the significant
technical contributions that he has already made. He finds it difficult to decide what he
should choose for his third topic, partly because he believes that, as usual, the faculty
will select the first topic on the list anyway. As the third one, he offers a rather vague
natural philosophical theme. To his surprise and consternation, the faculty chooses that
one. Instead of now familiarizing himself with the state of art of the discipline and, in
particular the really significant prior discovery that shook the entire field, he immerses
himself in the work of a rather obscure philosopher. But his lecture then penetrates as
deeplyasneverbeforeintoafieldthathadoccupiedandchallengedthegreatestthinkersof
mankindsinceclassicalantiquity,anditevenhintsatthegreatestdiscoveryofthephysics
of the followingcentury.Eventhe contributionofthe superstar ofGermanscience,who
hadindependentlyapproachedthesamesubjectfromadifferentpointofview,fadedinto
insignificanceincomparisonwiththedepthofinsightofouryoungmathematician.Other
famous scientists entered the stage with grotesque errors of judgment on the topic and
contentofthehabilitationlectureafterithadbeenpublishedbyafriendaftertheuntimely
deathofourhero.Subsequentgenerationsofmathematiciansworkedouttheideasoutlined
inthebrieflectureandconfirmedtheirfullvalidityandsoundnessandextraordinaryrange
andpotential.
However,thisisnotanovel,becausesomethingsimilardidactuallyoccur.Wehopeand
trustthatreadersforgivetheauthorcertainexaggerations,and,ofcourse,inthefollowing
pages,everythingwillbe representedcorrectly.Theyoungmathematicianwas Bernhard
Riemann,andthe lecturewasentitled “UeberdieHypothesen,welchederGeometriezu
Grundeliegen”(Onthehypotheseswhichlieatthebasesofgeometry).Themathematical
genius was Carl Friedrich Gauss, the scientific superstar Hermann von Helmholtz, the
fundamentalprior mathematical discovery non-Euclideangeometry, the philosopher the
v
vi Preface
now forgotten Johann Friedrich Herbart, the discovery in physics the theory of general
relativityofAlbertEinstein.Thepeoplewhocameupwithcompletelywrongjudgements
included the psychologist Wilhelm Wundt and the philosopher Bertrand Russell. The
friend who took care of the posthumous publication was the mathematician Richard
Dedekind.ThegenerationsofsubsequentmathematiciansforwhoseresearchRiemann’s
ideaswereamajorinspirationincludetheauthoroftheselines.
Typically,scientistsreadascientifictextfromthepointofviewofthecurrentstateof
science, interpret it in terms of subsequent developments, and seek at best unexplored
potential for current scientific problems. Historians, however, want to determine the
position of a text within the discourse of its time, reconstruct its origin and analyze the
history of its reception. Although in the current debate on the role of the humanities,
the importance of the historical sciences for understanding the present is emphasized,
mathematiciansareinterestedinthetimelesscontentandnotinthehistoricalcontingencies
of scientific texts. Scientific projects that proved futile are either of no interest to the
scientist or constitute annoying obstacles on a path that could have been straighter. For
historians,incontrast,theycanprovideimportantinsightsintothehistoryofideasandthe
dynamicsofdiscourses.Forthescientist,textswhoseeffecthasfadedarewithoutinterest.
Forthehistorianthislossofinterestispartofthereceptionhistory.
This edition of Riemann’s “Ueber die Hypothesen, welche der Geometrie zugrunde
liegen” tries to accept these challenges. The publisher is a professional scientist, not a
historianofscience.Therefore,thehistoryisalsosometimesreadbackwards.Inparticular,
forthiseditionnothoroughphilologicalstudieshavebeencarriedout.
Although there will also be a formal mathematical chapter, I shall mainly attempt to
explain the basic concepts and basic ideas in words, even if this will inevitably lead to
somelossofprecision.
Thecurrentbookis asomewhatexpandedEnglishtranslationofmyoriginalGerman
work.Inparticular,insteadofalsotranslatingHermannWeyl’smathematicalcommentary
thathadbeenincludedintheGermanversion,Ihavewrittenamoredetailedmathematical
section.ThatsectionprovidesthemathematicalbackgroundandputsRiemann’sreasoning
intothemoregeneralandsystematicperspectiveachievedbyhisfollowersonthebasisof
hisseminalideas.Readerswhoarenotsomuchinterestedinmathematicaldetailsmayskip
thissection,sinceinanothersection,IhavealsoexplainedRiemann’sreasoningverbally
asanalternativetomathematicaldeductions.
As mentioned before, I am not a historian of mathematics. Therefore, I am much
indebted to some historians of mathematics, namely Erhard Scholz, Rüdiger Thiele and
KlausVolkert,fortheirveryusefulcomments,corrections,suggestionsandreferences.Of
course,theresponsibilityforanyshortcomingsrestswithmealone.Iamalsogratefulto
theHelmholtzexpertJochenBrüningforhisinsightfulcomments.
IthankIngoBrüggemann,thelibrarianoftheMaxPlanckInstituteforMathematicsin
the Sciences,andhisstaff fortheirvaluableandefficientassistance in the acquisitionof
literature.
Preface vii
The largest amount of gratitude I owe to my friend, the late Olaf Breidbach, for his
initiativeinestablishingtheseriesinwhichthiseditionofRiemann’stextmaynowappear,
as well as for the many discussions on a wide range of scientific topics over several
decades.Afterhisuntimelydeath,IamnowaloneinchargeofthisseriesofClassicTexts
intheSciencesanditsGermanlanguagecounterpart,whichwehadfoundedtogetherwith
so much enthusiasm.I hope thatI shall be able to preservehis spirit as one of the great
historiansofscienceofourtimes.
Leipzig,Germany JürgenJost
August2015
Contents
1 Introduction ......................................................................... 1
2 HistoricalIntroduction............................................................. 9
2.1 TheSpaceProbleminPhysics,fromAristotletoNewton................... 9
2.2 Kant’sPhilosophyofSpace ................................................... 17
2.3 EuclideanSpaceastheBasicModel.......................................... 21
2.4 TheDevelopmentofGeometry:Non-EuclidianandDifferential
Geometry....................................................................... 25
2.5 TheStoryofRiemann’sHabilitationAddress................................ 27
3 Riemann’sText...................................................................... 29
4 PresentationoftheText ............................................................ 43
4.1 ShortSummary................................................................. 43
4.2 TheMainResultsoftheText.................................................. 44
4.3 Riemann’sReasoning.......................................................... 45
4.4 MathematicalCommentary.................................................... 62
4.4.1 TheConceptofaManifold .......................................... 63
4.4.2 TensorCalculus....................................................... 67
4.4.3 MetricStructures..................................................... 72
4.4.4 GeodesicCurves...................................................... 74
4.4.5 NormalCoordinates.................................................. 79
4.4.6 Riemann’sAbstractReasoning...................................... 86
4.4.7 FlatnessandCurvature............................................... 88
4.4.8 SubmanifoldsofEuclideanSpace................................... 101
4.4.9 SectionalCurvature .................................................. 104
4.4.10 SpacesofConstantCurvature ....................................... 112
4.5 GoingThroughRiemann’sText............................................... 115
ix
