1670:
254:). Furthermore, given an oriented surface, two Riemannian metrics induce the same holomorphic atlas if and only if they are conformal to one another. For this reason, the study of Riemann surfaces is identical to the study of conformal classes of Riemannian metrics on oriented surfaces.
1109:
788:
245:
coordinate atlas consisting of isothermal coordinate charts may be viewed as a holomorphic coordinate atlas. This demonstrates that a
Riemannian metric and an orientation on a two-dimensional manifold combine to induce the structure of a
1196:
1542:
545:
129:
421:
273:, among others. In this context, it is natural to investigate the existence of generalized solutions, which satisfy the relevant partial differential equations but are no longer interpretable as
1238:
2158:[General solution of the problem of mapping the parts of a given surface on another given surface in such a way that the mapping resembles what is depicted in the smallest parts]. In
648:
2156:"Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Flache auf einer andern gegebnen FlÀche so abzubilden, dass die Abbildung dem Abgebildeten in den kleinsten Theilen Àhnlich wird"
904:
950:
656:
614:
1382:
1309:
568:
152:
1332:
942:
459:
1530:
823:
1131:
588:
1490:
1467:
1428:
1405:
2622:
2409:
2013:
1665:{\displaystyle K=-{\frac {1}{2}}e^{-\rho }\left({\frac {\partial ^{2}\rho }{\partial u^{2}}}+{\frac {\partial ^{2}\rho }{\partial v^{2}}}\right).}
237:, is necessarily angle-preserving. The angle-preserving property together with orientation-preservation is one characterization (among many) of
2385:
Bulletin
International de l'Académie des Sciences de Cracovie: Classe des Sciences Mathématiques et Naturelles. Série A: Sciences Mathématiques
1492:
implies the diagonality of the metric, and the norm-preserving property of the Hodge star implies the equality of the two diagonal components.
158:. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.)
220:
found in 1916 the general existence of isothermal coordinates for
Riemannian metrics of lower regularity, including smooth metrics and even
1958:
467:
208:, so that his method is fundamentally restricted to the real-analytic context. Following innovations in the theory of two-dimensional
2627:
2381:"Zur Theorie der konformen Abbildung. Konforme Abbildung nichtanalytischer, singularitĂ€tenfreier FlĂ€chenstĂŒcke auf ebene Gebiete"
54:
1253:
282:
2556:
2331:
2194:
2110:
165:. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold.
330:
1731:
168:
By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually
2486:
1248:
The existence of isothermal coordinates on a smooth two-dimensional
Riemannian manifold is a corollary of the standard
1204:
286:
619:
2506:
2452:
2265:
1994:
1928:
1104:{\displaystyle e^{\rho }\,|dw|^{2}=e^{\rho }|w_{z}|^{2}|\,dz+{w_{\overline {z}} \over w_{z}}\,d{\overline {z}}|^{2},}
831:
783:{\displaystyle \lambda ={1 \over 4}(E+G+2{\sqrt {EG-F^{2}}}),\,\,\,{\displaystyle \mu ={(E-G+2iF) \over 4\lambda }}.}
2540:
2315:
1978:
2617:
209:
2143:, London Mathematical Society Lecture Note Series, vol. 274, Cambridge University Press, pp. 307â324
2600:
2490:
2159:
1986:
1912:
205:
2590:
2595:
2178:
306:
2295:
2539:. Applied Mathematical Sciences. Vol. 115 (Second edition of 1996 original ed.). New York:
2103:
1431:
593:
301:
The existence of isothermal coordinates can be proved by applying known existence theorems for the
169:
1349:
1201:
has a diffeomorphic solution. Such a solution has been proved to exist in any neighbourhood where
2363:
1686:
1278:
Isothermal coordinates are constructed from such a function in the following way. Harmonicity of
2400:
1285:
1256:. In the present context, the relevant elliptic equation is the condition for a function to be
201:
1243:
553:
257:
By the 1950s, expositions of the ideas of Korn and
Lichtenstein were put into the language of
137:
2483:
Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials
1191:{\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}}
21:
1317:
912:
429:
2566:
2516:
2462:
2359:
2275:
2222:
2151:
2120:
2086:
1938:
1904:
1503:
1430:
are orthogonal to one another and hence linearly independent, and it then follows from the
796:
238:
189:
162:
2574:
2524:
2470:
2392:
2283:
2230:
2128:
573:
221:
8:
2221:. Source Books in the History of the Sciences. Translated by Evans, Herbert P. New York:
1962:
1312:
29:
2052:
1472:
1449:
1410:
1387:
310:
176:
vanishes. In dimensions > 3, a metric is locally conformally flat if and only if its
2428:
2299:
2214:
2032:
1954:
1691:
1533:
258:
37:
172:. In dimension 3, a Riemannian metric is locally conformally flat if and only if its
2552:
2532:
2502:
2478:
2448:
2376:
2327:
2307:
2261:
2210:
2190:
2106:
2095:
2074:
1990:
1924:
1269:
1257:
302:
270:
262:
217:
193:
45:
1335:
1260:
relative to the
Riemannian metric. The local solvability then states that any point
2570:
2544:
2520:
2494:
2466:
2418:
2388:
2319:
2279:
2253:
2226:
2182:
2124:
2064:
2022:
1916:
274:
251:
41:
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2512:
2458:
2380:
2345:
2341:
2323:
2271:
2249:
2186:
2155:
2116:
2082:
1934:
1908:
1446:. This coordinate system is automatically isothermal, since the orthogonality of
247:
197:
155:
2440:
2350:
Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Berlin
2044:
1896:
317:. A simpler approach to the Beltrami equation has been given more recently by
314:
278:
228:
2548:
2257:
192:
proved the existence of isothermal coordinates on an arbitrary surface with a
2611:
2303:
2136:
2102:(Revised and updated second edition of 1976 original ed.). Mineola, NY:
2078:
1681:
318:
231:
between isothermal coordinate charts, which is a map between open subsets of
173:
33:
2009:"An elementary proof of the existence of isothermal parameters on a surface"
2048:
1888:
1872:
1244:
Existence via local solvability for elliptic partial differential equations
290:
2405:"On the solutions of quasi-linear elliptic partial differential equations"
2291:
2004:
1970:
1946:
1895:. University Lecture Series. Vol. 38. With supplemental chapters by
266:
213:
177:
17:
2294:(1914). "Zwei Anwendungen der Methode der sukzessiven AnnÀherungen". In
2069:
1920:
2498:
2432:
2036:
1974:
1900:
1755:
2423:
2404:
2027:
2008:
2447:(Third edition of 1975 original ed.). Publish or Perish, Inc.
2445:
A comprehensive introduction to differential geometry. Volume four
540:{\displaystyle ds^{2}=\lambda |\,dz+\mu \,d{\overline {z}}|^{2},}
2141:
Le théorÚme d'intégrabilité des structures presque complexes.
1719:
227:
Given a
Riemannian metric on a two-dimensional manifold, the
1911:(Second edition of 1966 original ed.). Providence, RI:
1875:(1952), "Conformality with respect to Riemannian metrics.",
1703:
161:
Isothermal coordinates on surfaces were first introduced by
1282:
is identical to the closedness of the differential 1-form
124:{\displaystyle g=\varphi (dx_{1}^{2}+\cdots +dx_{n}^{2}),}
2177:. Cambridge Library Collection (in German). New York:
416:{\displaystyle ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},}
1545:
1506:
1475:
1452:
1413:
1390:
1352:
1320:
1288:
1207:
1134:
953:
915:
834:
799:
727:
659:
622:
596:
576:
556:
470:
432:
333:
183:
140:
57:
2057:
2312:Mathematische Abhandlungen Hermann Amandus Schwarz
1664:
1524:
1484:
1461:
1422:
1399:
1376:
1326:
1303:
1232:
1190:
1103:
936:
898:
817:
782:
642:
608:
582:
562:
539:
453:
415:
146:
123:
2410:Transactions of the American Mathematical Society
2243:
2166:. Altona: Hammerich und Heineking. pp. 1â30.
2053:"Some regularity theorems in Riemannian geometry"
1985:. Lectures in Applied Mathematics. Vol. 3A.
1969:
1844:
1817:
1793:
1775:
1442:form a coordinate system on some neighborhood of
1233:{\displaystyle \lVert \mu \rVert _{\infty }<1}
285:on two-dimensional domains, leading later to the
204:. The construction used by Gauss made use of the
44:. This means that in isothermal coordinates, the
2609:
2014:Proceedings of the American Mathematical Society
643:{\displaystyle \left\vert \mu \right\vert <1}
196:Riemannian metric, following earlier results of
899:{\displaystyle ds^{2}=e^{\rho }(du^{2}+dv^{2})}
2537:Partial differential equations I. Basic theory
2346:"Sur la construction des cartes géographiques"
2100:Differential geometry of curves & surfaces
2043:
1856:
324:If the Riemannian metric is given locally as
281:in his seminal 1938 article on the theory of
2375:
1741:
1215:
1208:
2623:Coordinate systems in differential geometry
2135:
1959:Courant Institute of Mathematical Sciences
1829:
2422:
2068:
2026:
1338:thus implies the existence of a function
1334:associated to the Riemannian metric. The
1072:
1033:
964:
725:
724:
723:
508:
495:
396:
383:
376:
353:
2340:
2164:Astronomische Abhandlungen, Drittes Heft
2094:
1713:
277:in the usual way. This was initiated by
1887:
1871:
1805:
1769:
1254:elliptic partial differential equations
283:elliptic partial differential equations
2610:
2531:
2477:
2439:
2399:
2358:
1840:
1781:
1749:
1745:
1725:
909:with Ï smooth. The complex coordinate
2368:(in French). Paris: Gauthier-Villars.
2246:An introduction to TeichmĂŒller spaces
2208:
2172:
2150:
2003:
1953:. Notes taken by Rodlitz, Esther and
1765:
1709:
1495:
2290:
2244:Imayoshi, Y.; Taniguchi, M. (1992).
1945:
1761:
1737:
296:
2487:Mathematical Surveys and Monographs
1893:Lectures on quasiconformal mappings
1275:with nowhere-vanishing derivative.
13:
1638:
1624:
1601:
1587:
1219:
1179:
1171:
1146:
1138:
287:measurable Riemann mapping theorem
184:Isothermal coordinates on surfaces
14:
2639:
2583:
1384:By definition of the Hodge star,
2489:. Vol. 81. Providence, RI:
825:the metric should take the form
305:, which rely on L estimates for
32:are local coordinates where the
2173:Gauss, Carl Friedrich (2011) .
1850:
1845:Bers, John & Schechter 1979
1834:
426:then in the complex coordinate
2628:Partial differential equations
1983:Partial differential equations
1877:Ann. Acad. Sci. Fenn. Ser. A I
1823:
1811:
1799:
1787:
1519:
1507:
1500:In the isothermal coordinates
1088:
1029:
1018:
1002:
978:
966:
893:
861:
812:
800:
761:
737:
717:
676:
524:
491:
210:partial differential equations
115:
67:
1:
2491:American Mathematical Society
2211:"On conformal representation"
1987:American Mathematical Society
1913:American Mathematical Society
1864:
1818:Imayoshi & Taniguchi 1992
1794:Imayoshi & Taniguchi 1992
609:{\displaystyle \lambda >0}
2324:10.1007/978-3-642-50735-9_16
2219:A source book in mathematics
2187:10.1017/CBO9781139058254.005
1377:{\displaystyle dv=\star du.}
1154:
1122:) will be isothermal if the
1081:
1054:
517:
7:
2596:Encyclopedia of Mathematics
1748:, Addendum 1 to Chapter 9;
1675:
307:singular integral operators
10:
2644:
2365:Ćuvres de Lagrange: tome 4
2179:Cambridge University Press
1252:result in the analysis of
793:In isothermal coordinates
2549:10.1007/978-1-4419-7055-8
2258:10.1007/978-4-431-68174-8
2204:Translated to English in:
1857:DeTurck & Kazdan 1981
1304:{\displaystyle \star du,}
1114:so that the coordinates (
206:CauchyâKowalevski theorem
2591:"Isothermal coordinates"
2104:Dover Publications, Inc.
1697:
1432:inverse function theorem
563:{\displaystyle \lambda }
250:(i.e. a one-dimensional
170:locally conformally flat
147:{\displaystyle \varphi }
1536:takes the simpler form
200:in the special case of
2314:. Berlin, Heidelberg:
1830:Douady & Buff 2000
1666:
1526:
1486:
1463:
1424:
1401:
1378:
1328:
1327:{\displaystyle \star }
1305:
1234:
1192:
1105:
938:
937:{\displaystyle w=u+iv}
900:
819:
784:
644:
610:
584:
564:
541:
455:
454:{\displaystyle z=x+iy}
417:
202:surfaces of revolution
148:
125:
26:isothermal coordinates
2618:Differential geometry
2096:do Carmo, Manfredo P.
1667:
1527:
1525:{\displaystyle (u,v)}
1487:
1464:
1425:
1402:
1379:
1329:
1306:
1235:
1193:
1106:
939:
901:
820:
818:{\displaystyle (u,v)}
785:
645:
611:
585:
565:
542:
456:
418:
239:holomorphic functions
149:
126:
48:locally has the form
22:differential geometry
2318:. pp. 215â229.
2223:McGraw-Hill Book Co.
1843:, pp. 440â441;
1752:, Proposition 3.9.3.
1687:Liouville's equation
1543:
1504:
1473:
1450:
1411:
1388:
1350:
1318:
1286:
1268:on which there is a
1205:
1132:
951:
913:
832:
797:
657:
620:
594:
583:{\displaystyle \mu }
574:
554:
468:
461:, it takes the form
430:
331:
190:Carl Friedrich Gauss
138:
55:
2215:Smith, David Eugene
2139:; Buff, X. (2000),
2070:10.24033/asens.1405
1963:New York University
1313:Hodge star operator
1264:has a neighborhood
259:complex derivatives
229:transition function
114:
87:
30:Riemannian manifold
2533:Taylor, Michael E.
2479:Taylor, Michael E.
2401:Morrey, Charles B.
2377:Lichtenstein, LĂ©on
2225:pp. 463â475.
2045:DeTurck, Dennis M.
2005:Chern, Shiing-shen
1847:, pp. 228â230
1692:Quasiconformal map
1662:
1534:Gaussian curvature
1522:
1496:Gaussian curvature
1485:{\displaystyle dv}
1482:
1462:{\displaystyle du}
1459:
1423:{\displaystyle dv}
1420:
1400:{\displaystyle du}
1397:
1374:
1324:
1311:defined using the
1301:
1230:
1188:
1101:
934:
896:
815:
780:
775:
640:
606:
580:
560:
537:
451:
413:
144:
121:
100:
73:
20:, specifically in
2558:978-1-4419-7054-1
2333:978-3-642-50426-6
2196:978-1-108-03226-1
2160:Schumacher, H. C.
2112:978-0-486-80699-0
1979:Schechter, Martin
1965:. pp. 15â35.
1921:10.1090/ulect/038
1820:, pp. 92â104
1808:, pp. 85â115
1742:Lichtenstein 1916
1652:
1615:
1563:
1270:harmonic function
1250:local solvability
1186:
1160:
1157:
1124:Beltrami equation
1084:
1070:
1057:
773:
715:
674:
520:
303:Beltrami equation
297:Beltrami equation
275:coordinate charts
271:Shiing-shen Chern
263:Beltrami equation
222:Hölder continuous
218:Leon Lichtenstein
46:Riemannian metric
2635:
2604:
2578:
2528:
2499:10.1090/surv/081
2474:
2436:
2426:
2396:
2369:
2353:
2337:
2308:Lichtenstein, L.
2296:Carathéodory, C.
2287:
2234:
2200:
2167:
2144:
2132:
2090:
2072:
2049:Kazdan, Jerry L.
2040:
2030:
2000:
1966:
1955:Pollack, Richard
1951:Riemann surfaces
1942:
1889:Ahlfors, Lars V.
1884:
1873:Ahlfors, Lars V.
1859:
1854:
1848:
1838:
1832:
1827:
1821:
1815:
1809:
1803:
1797:
1796:, pp. 20â21
1791:
1785:
1779:
1773:
1759:
1753:
1735:
1729:
1723:
1717:
1707:
1671:
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1564:
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1488:
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1398:
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1310:
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1281:
1274:
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1263:
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764:
735:
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714:
713:
695:
675:
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649:
647:
646:
641:
633:
615:
613:
612:
607:
590:are smooth with
589:
587:
586:
581:
569:
567:
566:
561:
546:
544:
543:
538:
533:
532:
527:
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483:
482:
460:
458:
457:
452:
422:
420:
419:
414:
409:
408:
366:
365:
346:
345:
252:complex manifold
236:
153:
151:
150:
145:
130:
128:
127:
122:
113:
108:
86:
81:
42:Euclidean metric
2643:
2642:
2638:
2637:
2636:
2634:
2633:
2632:
2608:
2607:
2589:
2586:
2581:
2559:
2509:
2455:
2441:Spivak, Michael
2424:10.2307/1989904
2372:
2334:
2268:
2250:Springer-Verlag
2240:
2237:
2203:
2197:
2175:Werke: Volume 4
2113:
2028:10.2307/2032933
1997:
1931:
1867:
1862:
1855:
1851:
1839:
1835:
1828:
1824:
1816:
1812:
1804:
1800:
1792:
1788:
1780:
1776:
1760:
1756:
1736:
1732:
1728:, Theorem 9.18.
1724:
1720:
1708:
1704:
1700:
1678:
1645:
1641:
1637:
1627:
1623:
1622:
1620:
1608:
1604:
1600:
1590:
1586:
1585:
1583:
1582:
1578:
1569:
1565:
1555:
1544:
1541:
1540:
1505:
1502:
1501:
1498:
1474:
1471:
1470:
1451:
1448:
1447:
1443:
1439:
1435:
1412:
1409:
1408:
1389:
1386:
1385:
1351:
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1347:
1343:
1339:
1319:
1316:
1315:
1287:
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1283:
1279:
1272:
1265:
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1246:
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1214:
1206:
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1202:
1178:
1170:
1168:
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1137:
1135:
1133:
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1129:
1092:
1087:
1086:
1076:
1064:
1060:
1049:
1045:
1043:
1028:
1022:
1017:
1016:
1010:
1006:
1001:
995:
991:
982:
977:
976:
965:
958:
954:
952:
949:
948:
914:
911:
910:
887:
883:
871:
867:
855:
851:
842:
838:
833:
830:
829:
798:
795:
794:
765:
736:
734:
726:
709:
705:
694:
666:
658:
655:
654:
623:
621:
618:
617:
595:
592:
591:
575:
572:
571:
555:
552:
551:
528:
523:
522:
512:
490:
478:
474:
469:
466:
465:
431:
428:
427:
404:
400:
361:
357:
341:
337:
332:
329:
328:
299:
248:Riemann surface
232:
198:Joseph Lagrange
186:
156:smooth function
139:
136:
135:
109:
104:
82:
77:
56:
53:
52:
12:
11:
5:
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2584:External links
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2209:Gauss (1929).
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1715:
1714:Lagrange 1779
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1682:Conformal map
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2311:
2245:
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2174:
2163:
2152:Gauss, C. F.
2140:
2099:
2060:
2056:
2018:
2012:
1982:
1971:Bers, Lipman
1950:
1947:Bers, Lipman
1892:
1880:
1876:
1852:
1836:
1825:
1813:
1806:Ahlfors 2006
1801:
1789:
1777:
1770:Ahlfors 2006
1757:
1733:
1721:
1705:
1499:
1277:
1249:
1247:
1200:
1123:
1119:
1115:
1113:
908:
792:
549:
425:
323:
300:
291:Lars Ahlfors
256:
242:
241:, and so an
233:
226:
187:
167:
160:
133:
25:
15:
2387:: 192â217.
2059:. SĂ©rie 4.
1975:John, Fritz
1897:C. J. Earle
1841:Taylor 2011
1782:Morrey 1938
1750:Taylor 2000
1746:Spivak 1999
1726:Spivak 1999
267:Lipman Bers
214:Arthur Korn
180:vanishes.
178:Weyl tensor
18:mathematics
2612:Categories
2575:1206.35002
2525:0963.35211
2471:1213.53001
2393:46.0547.01
2352:: 161â210.
2304:Landau, E.
2284:0754.30001
2231:55.0583.01
2129:1352.53002
1865:References
1766:Chern 1955
1710:Gauss 1825
944:satisfies
650:. In fact
293:and Bers.
2601:EMS Press
2248:. Tokyo:
2079:0012-9593
1762:Bers 1958
1738:Korn 1914
1639:∂
1634:ρ
1625:∂
1602:∂
1597:ρ
1588:∂
1574:ρ
1571:−
1553:−
1363:⋆
1322:⋆
1290:⋆
1220:∞
1216:‖
1212:μ
1209:‖
1180:∂
1172:∂
1166:μ
1155:¯
1147:∂
1139:∂
1082:¯
1055:¯
997:ρ
960:ρ
857:ρ
770:λ
744:−
729:μ
703:−
661:λ
628:μ
598:λ
578:μ
558:λ
518:¯
506:μ
488:λ
224:metrics.
188:In 1822,
142:φ
92:⋯
65:φ
38:conformal
2541:Springer
2535:(2011).
2481:(2000).
2443:(1999).
2403:(1938).
2379:(1916).
2344:(1779).
2316:Springer
2310:(eds.).
2292:Korn, A.
2154:(1825).
2098:(2016).
2051:(1981).
2007:(1955).
1981:(1979).
1949:(1958).
1891:(2006).
1676:See also
1258:harmonic
311:CalderĂłn
261:and the
243:oriented
2603:, 2001
2567:2744150
2517:1766415
2463:0532833
2433:1989904
2276:1215481
2217:(ed.).
2162:(ed.).
2121:3837152
2087:0644518
2037:2032933
1939:2241787
315:Zygmund
40:to the
2573:
2565:
2555:
2523:
2515:
2505:
2469:
2461:
2451:
2431:
2391:
2330:
2282:
2274:
2264:
2229:
2193:
2127:
2119:
2109:
2085:
2077:
2035:
1993:
1937:
1927:
1901:I. Kra
1883:: 1â22
1532:, the
550:where
134:where
34:metric
2429:JSTOR
2213:. In
2033:JSTOR
1698:Notes
1434:that
1346:with
163:Gauss
28:on a
2553:ISBN
2503:ISBN
2449:ISBN
2328:ISBN
2262:ISBN
2191:ISBN
2107:ISBN
2075:ISSN
1991:ISBN
1925:ISBN
1907:and
1469:and
1438:and
1407:and
1225:<
635:<
616:and
601:>
570:and
313:and
269:and
2571:Zbl
2545:doi
2521:Zbl
2495:doi
2467:Zbl
2419:doi
2389:JFM
2320:doi
2280:Zbl
2254:doi
2227:JFM
2183:doi
2125:Zbl
2065:doi
2023:doi
1961:at
1917:doi
1881:206
1342:on
309:of
289:of
265:by
212:by
36:is
16:In
2614::
2599:,
2593:,
2569:.
2563:MR
2561:.
2551:.
2543:.
2519:.
2513:MR
2511:.
2501:.
2493:.
2485:.
2465:.
2459:MR
2457:.
2427:.
2415:43
2413:.
2407:.
2383:.
2348:.
2326:.
2306:;
2302:;
2298:;
2278:.
2272:MR
2270:.
2260:.
2252:.
2189:.
2181:.
2123:.
2117:MR
2115:.
2083:MR
2081:.
2073:.
2061:14
2055:.
2047:;
2031:.
2017:.
2011:.
1989:.
1977:;
1973:;
1957:.
1935:MR
1933:.
1923:.
1915:.
1903:,
1899:,
1879:,
1768:;
1764:;
1744:;
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2547::
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2497::
2473:.
2435:.
2421::
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2336:.
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2256::
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2199:.
2185::
2131:.
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2019:6
1999:.
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1919::
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62:=
59:g
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