From Riemann to differential geometry and relativity

664 Pages·2017·9.119 MB·English

by  Ji, Lizhen, Papadopoulos, Athanase, Yamada, Sumio (eds.)

Lizhen Ji
Athanase Papadopoulos
Sumio Yamada Editors
From Riemann to
Differential Geometry
and Relativity
From Riemann to Differential Geometry
and Relativity
Lizhen Ji Athanase Papadopoulos
(cid:129)
Sumio Yamada
Editors
From Riemann to Differential
Geometry and Relativity
123
Editors
Lizhen Ji Sumio Yamada
Department ofMathematics Department ofMathematics
University of Michigan Gakushuin University
AnnArbor, MI Tokyo
USA Japan
Athanase Papadopoulos
University of Strasbourg, CNRS
Institut deRecherche Mathématique
Avancée
StrasbourgCedex
France
ISBN978-3-319-60038-3 ISBN978-3-319-60039-0 (eBook)
DOI 10.1007/978-3-319-60039-0
LibraryofCongressControlNumber:2017942965
©SpringerInternationalPublishingAG2017
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Preface
Bernhard Riemann is one of those few mathematicians whose work made a pro-
found transformation of mathematics and physics. Not only his results are
far-reaching, but his vision and approach to mathematics were directly felt and
appreciated by all the later generations of mathematicians.
To say something original on Riemann’s work is not easy, not because every-
thing about him is known—far from it, but because it requires a profound reading
and understanding of his mathematical writings, which are difficult, involving
hidden geometric arguments, sometimes originating in physics and most of all
relying on his broad intuitive vision. Besides a familiarity with the mathematical
concepts involved, a reader of Riemann’s works must be capable offollowing his
verytersestyle.Anyonewhohasreadhishabilitationlecture,ÜberdieHypothesen,
welchederGeometriezuGrundeliegen,hasfeltitsunusualtone.Themathematical
ideas are expressed there in a broad and unusual language, and the results are
generallystatedwithoutsupportingproofsorcalculations.Furthermore,theseideas
are intertwined with philosophical and historical considerations, which may be
incomprehensibletoa reader who isnotsensible tohistory and philosophy. André
WeilmentionsthismemoirinaletterhewrotetohissisteronMarch26,1940,and
published in his Collected Papers (Springer Verlag, New York, Vol. 1, p. 244–
255).Hewritesthefollowing,talkingaboutalgebraicfunctionsofonevariable:“It
is generally believed that there is nothing left to do on algebraic functions of one
variable,becauseRiemann,whodiscoveredalmosteverythingweknowaboutthese
functions (I am excepting the works of Poincaré and Klein on uniformization, and
thoseofHurwitzandSeverioncorrespondences)didnotleaveforusanystatement
of a big problem that concerns them. I am without doubt one of the most knowl-
edgeablepersonsonthissubject;certainlybecauseIhadthegoodfortune(in1923)
tolearnitdirectlyfromRiemann’swritings,whosememoirisofcourseoneofthe
greatest things that a mathematician has ever written; there is not a single word
there that is not of consequence.”
Today, 150 years after Riemann’s death, some of his highly original ideas are
still poorly known to the mathematical community, in spite of the fact that a large
numberof books and articles were published on his work. The reason isthat these
v
vi Preface
books often concentrate on the results that are considered to lead to important
developments, leaving in the dark some of Riemann’s beautiful ideas that deserve
to be contemplated and further exploited. Actualizing these ideas and including
them in the context of current mathematics is a permanent necessity.
Several essays included in the present volume are the result of reading
Riemann’swritings,andtheothersaremotivatedbyhisideasastheyappearinthe
scientific literature.
The decision of editing this book was taken after two conferences held in
Strasbourg, the first one on June 12–14, 2014, whose subject was “Riemann,
topology and physics,” and the second one on September 18–20, 2014, whose
subject was “Riemann, Einstein and geometry,” where Riemann’s influence on
relativitytheorywasemphasized.Consequently,thisbookcontainsseveralchapters
on the latter theory.
Despite the variety of topics contained in this volume, there is one simple and
common purpose, to highlight—hopefully in a new way—some of Riemann’s
original ideas and their subsequent development.
We would like to take this opportunity to thank Elena Griniari from Springer
Verlagforherinterest,supportandefficienthelpinthisedition,andManfredKarbe
for his invaluable advice.
Editing such a book required hard work. We consider it an expression of our
gratitude for all that Riemann gave to human knowledge. His ghostly voice still
inspires us all.
Ann Arbor, USA Lizhen Ji
Strasbourg, France and Providence, USA Athanase Papadopoulos
Tokyo, Japan Sumio Yamada
April 2017
BernardRiemann(CourtesyofthelibraryoftheUniversityofGötttingen)
Riemann'swifeanddaughter(CourtesyofthelibraryoftheUniversityofGötttingen)
Contents
Preamble . .... .... .... .... ..... .... .... .... .... .... ..... ....
1 Looking Backward: From Euler to Riemann . .... .... ..... .... 1
Athanase Papadopoulos
1 Introduction ... .... ..... .... .... .... .... .... ..... .... 2
2 Functions . .... .... ..... .... .... .... .... .... ..... .... 10
3 Elliptic Integrals.... ..... .... .... .... .... .... ..... .... 20
4 Abelian Functions... ..... .... .... .... .... .... ..... .... 30
5 Hypergeometric Series .... .... .... .... .... .... ..... .... 32
6 The Zeta Function .. ..... .... .... .... .... .... ..... .... 33
7 On Space . .... .... ..... .... .... .... .... .... ..... .... 40
8 Topology . .... .... ..... .... .... .... .... .... ..... .... 46
9 Differential Geometry..... .... .... .... .... .... ..... .... 60
10 Trigonometric Series. ..... .... .... .... .... .... ..... .... 66
11 Integration .... .... ..... .... .... .... .... .... ..... .... 76
12 Conclusion .... .... ..... .... .... .... .... .... ..... .... 78
References. .... .... .... ..... .... .... .... .... .... ..... .... 81
Part I Mathematics and Physics
2 Riemann on Geometry, Physics, and Philosophy—Some
Remarks.. .... .... .... ..... .... .... .... .... .... ..... .... 97
Jeremy Gray
1 Introduction ... .... ..... .... .... .... .... .... ..... .... 97
2 The Hypotheses .... ..... .... .... .... .... .... ..... .... 98
3 Influences. .... .... ..... .... .... .... .... .... ..... .... 101
4 Heat Diffusion and the Commentatio . .... .... .... ..... .... 106
References. .... .... .... ..... .... .... .... .... .... ..... .... 108
ix
x Contents
3 Some Remarks on “A Contribution to Electrodynamics” by
Bernhard Riemann . .... ..... .... .... .... .... .... ..... .... 111
Hubert Goenner
1 Introduction ... .... ..... .... .... .... .... .... ..... .... 111
2 Riemann’s New Result of 1858: The Retarded Potential ... .... 112
3 Gauss, Weber, and Riemann on Electrodynamic Interaction. .... 114
4 Riemann’s Paper.... ..... .... .... .... .... .... ..... .... 117
5 Concluding Remarks ..... .... .... .... .... .... ..... .... 119
References. .... .... .... ..... .... .... .... .... .... ..... .... 122
4 Riemann’s Memoir Über das Verschwinden der #-Functionen........ 125
Christian Houzel
1 Jacobi’s Inversion Problem. .... .... .... .... .... ..... .... 125
2 A Crucial Observation on Theta Functions. .... .... ..... .... 128
3 The First Step of Riemann’s Proof... .... .... .... ..... .... 129
4 The Second Step of Riemann’s Proof. .... .... .... ..... .... 130
5 The Conclusion of the Proof.... .... .... .... .... ..... .... 130
6 Later Developments . ..... .... .... .... .... .... ..... .... 133
References. .... .... .... ..... .... .... .... .... .... ..... .... 133
5 Riemann’s Work on Minimal Surfaces .. .... .... .... ..... .... 135
Sumio Yamada
1 Introduction ... .... ..... .... .... .... .... .... ..... .... 135
2 On the Surface of Least Area with a Given Boundary..... .... 136
3 Representation Formulas by Riemann and Weierstrass-Enneper..... 147
4 Closing Remarks ... ..... .... .... .... .... .... ..... .... 149
References. .... .... .... ..... .... .... .... .... .... ..... .... 150
6 Physics in Riemann’s Mathematical Papers .. .... .... ..... .... 151
Athanase Papadopoulos
1 Introduction ... .... ..... .... .... .... .... .... ..... .... 151
2 Function Theory and Riemann Surfaces... .... .... ..... .... 162
3 Riemann’s Memoir on Trigonometric Series.... .... ..... .... 173
4 Riemann’s Habilitationsvortrag 1854—Space and Matter... .... 181
5 The Commentatio and the Gleichgewicht der Electricität... .... 193
6 Riemann’s Other Papers... .... .... .... .... .... ..... .... 195
7 Conclusion .... .... ..... .... .... .... .... .... ..... .... 199
References. .... .... .... ..... .... .... .... .... .... ..... .... 199
7 Cauchy and Puiseux: Two Precursors of Riemann. .... ..... .... 209
Athanase Papadopoulos
1 Introduction ... .... ..... .... .... .... .... .... ..... .... 209
2 Algebraic Functions and Uniformization... .... .... ..... .... 210
3 Puiseux and Uniformization .... .... .... .... .... ..... .... 212