570:. Any such embedding can be scaled by an arbitrarily small constant so as to become short, relative to any given Riemannian metric on the surface. It follows from the NashâKuiper theorem that there are continuously differentiable isometric embeddings of any such Riemannian surface where the radius of a circumscribed ball is arbitrarily small. By contrast, no negatively curved closed surface can even be smoothly isometrically embedded in
558:
in the NashâKuiper theorem. For example, the image of any smooth isometric hypersurface immersion of the round sphere must itself be a round sphere. By contrast, the NashâKuiper theorem ensures the existence of continuously differentiable isometric hypersurface immersions of the round sphere which
378:
97:†â. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result.
958:
of the manifold. The proof of the global embedding theorem relies on Nash's implicit function theorem for isometric embeddings. This theorem has been generalized by a number of other authors to abstract contexts, where it is known as
629:. In analytical terms, the Euler equations have a formal similarity to the isometric embedding equations, via the quadratic nonlinearity in the first derivatives of the unknown function. The ideas of Nash's proof were abstracted by
580:) isometric embedding of an arbitrary closed Riemannian surface, there is a quantitative (positive) lower bound on the radius of a circumscribed ball in terms of the surface area and curvature of the embedded metric.
249:
888:
602:-dimensional Euclidean space. Although Whitney's theorem also applies to noncompact manifolds, such embeddings cannot simply be scaled by a small constant so as to become short. Nash proved that every
613:
At the time of Nash's work, his theorem was considered to be something of a mathematical curiosity. The result itself has not found major applications. However, Nash's method of proof was adapted by
511:
The isometric embeddings produced by the NashâKuiper theorem are often considered counterintuitive and pathological. They often fail to be smoothly differentiable. For example, a
120:
in the 1920s.) In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates. Nash's proof of the
413:, then there are more equations than unknowns. From this perspective, the existence of isometric embeddings given by the following theorem is considered surprising.
3020:
2211:
1915:
3015:
512:
2302:
2078:
1928:
971:
to make the Newton iteration converge: this is Newton's method with postconditioning. The fact that this technique furnishes a solution is in itself an
2326:
2521:
3073:
2108:
2083:
2391:
1970:
1875:
967:
to construct solutions. The standard Newton's method fails to converge when applied to the system; Nash uses smoothing operators defined by
373:{\displaystyle g_{ij}(x)=\sum _{\alpha =1}^{n}{\frac {\partial f^{\alpha }}{\partial x^{i}}}{\frac {\partial f^{\alpha }}{\partial x^{j}}}.}
2617:
1538:
470:. This map is not required to be isometric. Then there is a sequence of continuously differentiable isometric embeddings (or immersions)
2670:
2198:
2954:
1943:
1908:
1551:
1413:
1329:
1223:
554:. Furthermore, some smooth isometric embeddings exhibit rigidity phenomena which are violated by the largely unrestricted choice of
811:
2719:
1649:
642:
2311:
2702:
1309:
650:
1389:
630:
108:-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by
3068:
2914:
1313:
940:
236:
137:
3063:
2899:
2622:
2396:
2005:
1901:
1467:
1206:. Grundlehren der mathematischen Wissenschaften. Vol. 285. Translated from the Russian by A. B. SosinskiÄ. Berlin:
2944:
141:
74:
2949:
2919:
2627:
2583:
2564:
2331:
2275:
2026:
1544:
1801:
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2351:
2155:
1620:
1580:
1437:
951:
584:
2088:
2871:
2736:
2428:
2270:
1948:
1246:
646:
2568:
2538:
2462:
2452:
2408:
2238:
2191:
2134:
452:
2336:
2093:
960:
129:
2909:
2528:
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2243:
2165:
2160:
2098:
2031:
1990:
654:
1543:. Interscience Tracts in Pure and Applied Mathematics. Vol. 15. Reprinted in 1996. New YorkâLondon:
2558:
2553:
1814:
1809:
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181:
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2341:
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1995:
1858:. Applied Mathematical Sciences. Vol. 117 (Second edition of 1996 original ed.). New York:
1265:
946:
The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into
39:
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3000:
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2009:
1980:
1963:
1924:
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1432:
980:
976:
786:
725:
591:-dimensional Riemannian manifold admits an continuously differentiable isometric embedding into an
547:
482:
63:
43:
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2150:
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2016:
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964:
606:-dimensional Riemannian manifold admits a continuously differentiable isometric embedding into
2904:
2884:
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2697:
2511:
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2062:
1871:
1851:
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1241:
1219:
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614:
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528:
82:
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2231:
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2000:
1953:
1863:
1823:
1774:
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1589:
1491:
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1401:
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203:
59:
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1601:
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1503:
1458:
1423:
1381:
1339:
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1233:
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2894:
2724:
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1958:
1881:
1839:
1790:
1747:
1691:
1637:
1609:
1597:
1569:
1557:
1499:
1454:
1419:
1397:
1377:
1335:
1281:
1229:
1207:
767:
626:
540:
505:
86:
51:
559:
are arbitrarily close to (for instance) a topological embedding of the sphere as a small
1812:(1966). "Analyticity of the solutions of implicit function problem with analytic data".
1269:
2851:
2776:
2746:
2644:
2637:
2577:
2548:
2418:
2413:
2374:
1534:
798:
622:
618:
113:
1633:
1593:
3057:
3037:
2861:
2856:
2841:
2831:
2781:
2758:
2632:
2592:
2533:
2481:
2280:
1985:
1511:
771:
516:
117:
62:. For instance, bending but neither stretching nor tearing a page of paper gives an
1703:
2964:
2959:
2801:
2768:
2741:
2649:
2290:
1293:
17:
66:
of the page into
Euclidean space because curves drawn on the page retain the same
1893:
27:
Every
Riemannian manifold can be isometrically embedded into some Euclidean space
2807:
2796:
2753:
2654:
2255:
1686:
1657:
1495:
1195:
968:
963:. The basic idea in the proof of Nash's implicit function theorem is the use of
638:
125:
3032:
2990:
2816:
2729:
2361:
2265:
2103:
1867:
1405:
1277:
1215:
757:
617:
and Låszló Székelyhidi to construct low-regularity solutions, with prescribed
535:
cannot be smoothly isometrically immersed as a hypersurface, and a theorem of
1658:"Convex integration for Lipschitz mappings and counterexamples to regularity"
1515:
1450:
1396:. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 9. Berlin:
668:
The technical statement appearing in Nash's original paper is as follows: if
566:
Any closed and oriented two-dimensional manifold can be smoothly embedded in
2846:
2811:
2516:
2403:
741:
560:
493:
The theorem was originally proved by John Nash with the stronger assumption
448:
67:
47:
3010:
3005:
2995:
2386:
2207:
1975:
955:
950:. A local embedding theorem is much simpler and can be proved using the
195:
55:
1244:; Székelyhidi, Låszló Jr. (2013). "Dissipative continuous Euler flows".
132:. A simpler proof of the second Nash embedding theorem was obtained by
2176:
1835:
1786:
1743:
1373:
1321:
2602:
1676:
1827:
1778:
1735:
1365:
1486:
1260:
202:. In analytical terms, this may be viewed (relative to a smooth
2180:
1897:
1350:; Jacobowitz, Howard (1971). "Analytic isometric embeddings".
653:, constructing minimizers of minimal differentiability in the
883:{\displaystyle \langle u,v\rangle =df_{p}(u)\cdot df_{p}(v)}
1763:(1956). "The imbedding problem for Riemannian manifolds".
1856:
Partial differential equations III. Nonlinear equations
676:-dimensional Riemannian manifold (analytic or of class
814:
252:
1138:
975:
and of independent interest. In other contexts, the
81:) embeddings and the second for embeddings that are
2978:
2937:
2870:
2767:
2663:
2610:
2601:
2437:
2360:
2299:
2219:
2143:
2117:
2071:
2040:
1936:
1010:
882:
372:
587:, the NashâKuiper theorem shows that any closed
112:. (A local version of this result was proved by
109:
1435:[On the embedding theorem of J. Nash].
1114:
1102:
1090:
1078:
1066:
1054:
523:cannot be smoothly isometrically immersed into
2192:
1909:
550:cannot be smoothly isometrically immersed in
8:
1540:Foundations of differential geometry. Vol II
1166:
827:
815:
977:convergence of the standard Newton's method
2607:
2199:
2185:
2177:
2079:Fundamental theorem of Riemannian geometry
1916:
1902:
1894:
1685:
1675:
1485:
1259:
865:
840:
813:
583:In higher dimension, as follows from the
358:
343:
333:
324:
309:
299:
293:
282:
257:
251:
1621:Indagationes Mathematicae (Proceedings)
1581:Indagationes Mathematicae (Proceedings)
1042:
1038:
991:
939:, this is an underdetermined system of
133:
124:- case was later extrapolated into the
1154:
1014:
998:
1142:
546:-dimensional manifold of nonpositive
436:-dimensional Riemannian manifold and
58:means preserving the length of every
7:
1178:
1126:
1026:
574:. Moreover, for any smooth (or even
140:to an elliptic system, to which the
130:NashâMoser implicit function theorem
785:which is compatible with the given
104:theorem was published in 1954, the
724:is a non-compact manifold) and an
351:
336:
317:
302:
25:
1468:"A proof of Onsager's conjecture"
1433:"Zum Einbettungssatz von J. Nash"
684:†â), then there exists a number
243:unknown (real-valued) functions:
180:is a continuously differentiable
164:-dimensional Riemannian manifold
136:who reduced the set of nonlinear
1312:. Vol. 48. Providence, RI:
1139:De Lellis & Székelyhidi 2013
1093:, Corollary VII.5.4 and Note 15.
704:is a compact manifold, and with
3074:Theorems in Riemannian geometry
1310:Graduate Studies in Mathematics
1306:Introduction to the h-principle
1011:Eliashberg & Mishachev 2002
625:from the mathematical study of
198:of the Euclidean metric equals
2239:Differentiable/Smooth manifold
1394:Partial differential relations
941:partial differential equations
877:
871:
852:
846:
748:manifolds and for every point
593:arbitrarily small neighborhood
539:and Kuiper even says that any
272:
266:
237:partial differential equations
138:partial differential equations
110:Greene & Jacobowitz (1971)
1:
1634:10.1016/S1385-7258(55)50093-X
1594:10.1016/S1385-7258(55)50075-8
1314:American Mathematical Society
508:to obtain the theorem above.
504:. His method was modified by
2006:Raising and lowering indices
1618:-isometric imbeddings. II".
1188:General and cited references
651:Hilbert's nineteenth problem
2945:Classification of manifolds
1687:10.4007/annals.2003.157.715
1578:-isometric imbeddings. I".
1545:John Wiley & Sons, Inc.
1496:10.4007/annals.2018.188.3.4
1115:Burago & Zalgaller 1988
1103:Kobayashi & Nomizu 1969
1091:Kobayashi & Nomizu 1969
1079:Kobayashi & Nomizu 1969
1067:Kobayashi & Nomizu 1969
1055:Kobayashi & Nomizu 1969
979:had earlier been proved by
736:(also analytic or of class
142:contraction mapping theorem
75:continuously differentiable
3090:
2027:Pseudo-Riemannian manifold
1431:GĂŒnther, Matthias (1989).
954:of advanced calculus in a
70:however the page is bent.
3021:over commutative algebras
2156:Geometrization conjecture
1868:10.1007/978-1-4419-7049-7
1438:Mathematische Nachrichten
1406:10.1007/978-3-662-02267-2
1278:10.1007/s00222-012-0429-9
1216:10.1007/978-3-662-07441-1
952:implicit function theorem
585:Whitney embedding theorem
73:The first theorem is for
2737:Riemann curvature tensor
1451:10.1002/mana.19891440113
1304:; Mishachev, N. (2002).
1247:Inventiones Mathematicae
1167:MĂŒller & Ć verĂĄk 2003
805:in the following sense:
1720:isometric imbeddings".
956:coordinate neighborhood
637:, with a corresponding
455:) into Euclidean space
32:Nash embedding theorems
2529:Manifold with boundary
2244:Differential structure
2166:Uniformization theorem
2099:Nash embedding theorem
2032:Riemannian volume form
1991:Levi-Civita connection
1466:Isett, Philip (2018).
1204:Geometric inequalities
884:
655:calculus of variations
641:. This was applied by
491:
374:
298:
1815:Annals of Mathematics
1800:(Erratum:
1766:Annals of Mathematics
1723:Annals of Mathematics
1663:Annals of Mathematics
1473:Annals of Mathematics
1353:Annals of Mathematics
885:
451:smooth embedding (or
415:
375:
278:
182:topological embedding
150:NashâKuiper theorem (
46:can be isometrically
3069:Riemannian manifolds
2676:Covariant derivative
2227:Topological manifold
2089:GaussâBonnet theorem
1996:Covariant derivative
1531:Kobayashi, Shoshichi
1081:, Corollary VII.4.8.
812:
633:to the principle of
418:NashâKuiper theorem.
250:
40:John Forbes Nash Jr.
3064:Riemannian geometry
2710:Exterior derivative
2312:AtiyahâSinger index
2261:Riemannian manifold
2161:Poincaré conjecture
2022:Riemannian manifold
2010:Musical isomorphism
1925:Riemannian geometry
1610:Kuiper, Nicolaas H.
1570:Kuiper, Nicolaas H.
1270:2013InMat.193..377D
1129:, pp. 394â395.
1001:, pp. 147â151.
981:Leonid Kantorovitch
740:). That is Æ is an
726:isometric embedding
548:sectional curvature
178:isometric embedding
64:isometric embedding
44:Riemannian manifold
42:, state that every
18:NashâKuiper theorem
3016:Secondary calculus
2970:Singularity theory
2925:Parallel transport
2693:De Rham cohomology
2332:Generalized Stokes
2151:General relativity
2094:HopfâRinow theorem
2041:Types of manifolds
2017:Parallel transport
1852:Taylor, Michael E.
1242:De Lellis, Camillo
1117:, Corollary 6.2.2.
1105:, Theorem VII.5.6.
1069:, Theorem VII.5.3.
961:NashâMoser theorem
880:
635:convex integration
513:well-known theorem
483:converge uniformly
370:
156:embedding theorem)
144:could be applied.
36:imbedding theorems
3051:
3050:
2933:
2932:
2698:Differential form
2352:Whitney embedding
2286:Differential form
2174:
2173:
1877:978-1-4419-7048-0
1818:. Second Series.
1769:. Second Series.
1726:. Second Series.
1666:. Second Series.
1476:. Second Series.
1356:. Second Series.
1348:Greene, Robert E.
973:existence theorem
797:and the standard
664:embedding theorem
615:Camillo De Lellis
537:Shiing-Shen Chern
529:Einstein manifold
519:asserts that the
365:
331:
235:many first-order
209:) as a system of
16:(Redirected from
3081:
3043:Stratified space
3001:Fréchet manifold
2715:Interior product
2608:
2305:
2201:
2194:
2187:
2178:
1918:
1911:
1904:
1895:
1889:
1847:
1805:
1804:
1798:
1755:
1719:
1707:
1689:
1679:
1645:
1617:
1605:
1577:
1565:
1526:
1524:
1523:
1514:. Archived from
1489:
1462:
1427:
1385:
1343:
1297:
1263:
1237:
1200:Zalgaller, V. A.
1182:
1176:
1170:
1164:
1158:
1152:
1146:
1136:
1130:
1124:
1118:
1112:
1106:
1100:
1094:
1088:
1082:
1076:
1070:
1064:
1058:
1052:
1046:
1036:
1030:
1024:
1018:
1017:, Section 2.4.9.
1008:
1002:
996:
938:
929:
927:
926:
923:
920:
912:
893:for all vectors
889:
887:
886:
881:
870:
869:
845:
844:
609:
605:
601:
590:
579:
573:
569:
557:
553:
545:
533:scalar curvature
526:
521:hyperbolic plane
503:
488:
480:
476:
469:
458:
446:
435:
431:
412:
403:
401:
400:
397:
394:
386:
379:
377:
376:
371:
366:
364:
363:
362:
349:
348:
347:
334:
332:
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329:
328:
315:
314:
313:
300:
297:
292:
265:
264:
242:
234:
225:
223:
222:
219:
216:
208:
204:coordinate chart
201:
193:
175:
163:
155:
21:
3089:
3088:
3084:
3083:
3082:
3080:
3079:
3078:
3054:
3053:
3052:
3047:
2986:Banach manifold
2979:Generalizations
2974:
2929:
2866:
2763:
2725:Ricci curvature
2681:Cotangent space
2659:
2597:
2439:
2433:
2392:Exponential map
2356:
2301:
2295:
2215:
2205:
2175:
2170:
2139:
2118:Generalizations
2113:
2067:
2036:
1971:Exponential map
1932:
1922:
1892:
1878:
1850:
1828:10.2307/1970448
1808:
1799:
1779:10.2307/1969989
1759:
1758:
1736:10.2307/1969840
1715:
1710:
1648:
1613:
1608:
1573:
1568:
1554:
1535:Nomizu, Katsumi
1529:
1521:
1519:
1465:
1430:
1416:
1398:Springer-Verlag
1390:Gromov, Mikhael
1388:
1366:10.2307/1970760
1346:
1332:
1322:10.1090/gsm/048
1300:
1240:
1226:
1208:Springer-Verlag
1194:
1190:
1185:
1177:
1173:
1165:
1161:
1153:
1149:
1137:
1133:
1125:
1121:
1113:
1109:
1101:
1097:
1089:
1085:
1077:
1073:
1065:
1061:
1053:
1049:
1037:
1033:
1025:
1021:
1009:
1005:
997:
993:
989:
965:Newton's method
924:
921:
918:
917:
915:
914:
913:is larger than
910:
906:
861:
836:
810:
809:
794:
778:
765:
666:
647:VladimĂr Ć verĂĄk
627:fluid mechanics
623:Euler equations
607:
603:
596:
588:
575:
571:
567:
555:
551:
543:
524:
506:Nicolaas Kuiper
494:
486:
478:
471:
460:
456:
437:
433:
421:
398:
395:
392:
391:
389:
388:
384:
354:
350:
339:
335:
320:
316:
305:
301:
253:
248:
247:
240:
220:
217:
214:
213:
211:
210:
206:
199:
184:
165:
161:
158:
151:
52:Euclidean space
38:), named after
28:
23:
22:
15:
12:
11:
5:
3087:
3085:
3077:
3076:
3071:
3066:
3056:
3055:
3049:
3048:
3046:
3045:
3040:
3035:
3030:
3025:
3024:
3023:
3013:
3008:
3003:
2998:
2993:
2988:
2982:
2980:
2976:
2975:
2973:
2972:
2967:
2962:
2957:
2952:
2947:
2941:
2939:
2935:
2934:
2931:
2930:
2928:
2927:
2922:
2917:
2912:
2907:
2902:
2897:
2892:
2887:
2882:
2876:
2874:
2868:
2867:
2865:
2864:
2859:
2854:
2849:
2844:
2839:
2834:
2824:
2819:
2814:
2804:
2799:
2794:
2789:
2784:
2779:
2773:
2771:
2765:
2764:
2762:
2761:
2756:
2751:
2750:
2749:
2739:
2734:
2733:
2732:
2722:
2717:
2712:
2707:
2706:
2705:
2695:
2690:
2689:
2688:
2678:
2673:
2667:
2665:
2661:
2660:
2658:
2657:
2652:
2647:
2642:
2641:
2640:
2630:
2625:
2620:
2614:
2612:
2605:
2599:
2598:
2596:
2595:
2590:
2580:
2575:
2561:
2556:
2551:
2546:
2541:
2539:Parallelizable
2536:
2531:
2526:
2525:
2524:
2514:
2509:
2504:
2499:
2494:
2489:
2484:
2479:
2474:
2469:
2459:
2449:
2443:
2441:
2435:
2434:
2432:
2431:
2426:
2421:
2419:Lie derivative
2416:
2414:Integral curve
2411:
2406:
2401:
2400:
2399:
2389:
2384:
2383:
2382:
2375:Diffeomorphism
2372:
2366:
2364:
2358:
2357:
2355:
2354:
2349:
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2308:
2306:
2297:
2296:
2294:
2293:
2288:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2252:
2251:
2246:
2236:
2235:
2234:
2223:
2221:
2220:Basic concepts
2217:
2216:
2206:
2204:
2203:
2196:
2189:
2181:
2172:
2171:
2169:
2168:
2163:
2158:
2153:
2147:
2145:
2141:
2140:
2138:
2137:
2135:Sub-Riemannian
2132:
2127:
2121:
2119:
2115:
2114:
2112:
2111:
2106:
2101:
2096:
2091:
2086:
2081:
2075:
2073:
2069:
2068:
2066:
2065:
2060:
2055:
2050:
2044:
2042:
2038:
2037:
2035:
2034:
2029:
2024:
2019:
2014:
2013:
2012:
2003:
1998:
1993:
1983:
1978:
1973:
1968:
1967:
1966:
1961:
1956:
1951:
1940:
1938:
1937:Basic concepts
1934:
1933:
1923:
1921:
1920:
1913:
1906:
1898:
1891:
1890:
1876:
1848:
1822:(3): 345â355.
1806:
1756:
1730:(3): 383â396.
1708:
1670:(3): 715â742.
1646:
1606:
1566:
1552:
1527:
1480:(3): 871â963.
1463:
1445:(1): 165â187.
1428:
1414:
1386:
1360:(1): 189â204.
1344:
1330:
1302:Eliashberg, Y.
1298:
1254:(2): 377â407.
1238:
1224:
1196:Burago, Yu. D.
1191:
1189:
1186:
1184:
1183:
1171:
1159:
1157:, Section 2.4.
1147:
1131:
1119:
1107:
1095:
1083:
1071:
1059:
1047:
1031:
1019:
1013:, Chapter 21;
1003:
990:
988:
985:
904:
891:
890:
879:
876:
873:
868:
864:
860:
857:
854:
851:
848:
843:
839:
835:
832:
829:
826:
823:
820:
817:
792:
776:
761:
665:
659:
631:Mikhael Gromov
619:kinetic energy
381:
380:
369:
361:
357:
353:
346:
342:
338:
327:
323:
319:
312:
308:
304:
296:
291:
288:
285:
281:
277:
274:
271:
268:
263:
260:
256:
194:such that the
157:
148:
134:GĂŒnther (1989)
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3086:
3075:
3072:
3070:
3067:
3065:
3062:
3061:
3059:
3044:
3041:
3039:
3038:Supermanifold
3036:
3034:
3031:
3029:
3026:
3022:
3019:
3018:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2983:
2981:
2977:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2942:
2940:
2936:
2926:
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2901:
2898:
2896:
2893:
2891:
2888:
2886:
2883:
2881:
2878:
2877:
2875:
2873:
2869:
2863:
2860:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2829:
2825:
2823:
2820:
2818:
2815:
2813:
2809:
2805:
2803:
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2783:
2780:
2778:
2775:
2774:
2772:
2770:
2766:
2760:
2759:Wedge product
2757:
2755:
2752:
2748:
2745:
2744:
2743:
2740:
2738:
2735:
2731:
2728:
2727:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
2704:
2703:Vector-valued
2701:
2700:
2699:
2696:
2694:
2691:
2687:
2684:
2683:
2682:
2679:
2677:
2674:
2672:
2669:
2668:
2666:
2662:
2656:
2653:
2651:
2648:
2646:
2643:
2639:
2636:
2635:
2634:
2633:Tangent space
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2615:
2613:
2609:
2606:
2604:
2600:
2594:
2591:
2589:
2585:
2581:
2579:
2576:
2574:
2570:
2566:
2562:
2560:
2557:
2555:
2552:
2550:
2547:
2545:
2542:
2540:
2537:
2535:
2532:
2530:
2527:
2523:
2520:
2519:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2464:
2460:
2458:
2454:
2450:
2448:
2445:
2444:
2442:
2436:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2405:
2402:
2398:
2397:in Lie theory
2395:
2394:
2393:
2390:
2388:
2385:
2381:
2378:
2377:
2376:
2373:
2371:
2368:
2367:
2365:
2363:
2359:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2309:
2307:
2304:
2300:Main results
2298:
2292:
2289:
2287:
2284:
2282:
2281:Tangent space
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2250:
2247:
2245:
2242:
2241:
2240:
2237:
2233:
2230:
2229:
2228:
2225:
2224:
2222:
2218:
2213:
2209:
2202:
2197:
2195:
2190:
2188:
2183:
2182:
2179:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2148:
2146:
2142:
2136:
2133:
2131:
2128:
2126:
2123:
2122:
2120:
2116:
2110:
2109:Schur's lemma
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2084:Gauss's lemma
2082:
2080:
2077:
2076:
2074:
2070:
2064:
2061:
2059:
2056:
2054:
2051:
2049:
2046:
2045:
2043:
2039:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2011:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1988:
1987:
1986:Metric tensor
1984:
1982:
1981:Inner product
1979:
1977:
1974:
1972:
1969:
1965:
1962:
1960:
1957:
1955:
1952:
1950:
1947:
1946:
1945:
1942:
1941:
1939:
1935:
1930:
1926:
1919:
1914:
1912:
1907:
1905:
1900:
1899:
1896:
1887:
1883:
1879:
1873:
1869:
1865:
1861:
1857:
1853:
1849:
1845:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1816:
1811:
1807:
1802:
1796:
1792:
1788:
1784:
1780:
1776:
1772:
1768:
1767:
1762:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1725:
1724:
1718:
1713:
1709:
1705:
1701:
1697:
1693:
1688:
1683:
1678:
1673:
1669:
1665:
1664:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1622:
1616:
1612:(1955b). "On
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1582:
1576:
1572:(1955a). "On
1571:
1567:
1563:
1559:
1555:
1553:0-471-15732-5
1549:
1546:
1542:
1541:
1536:
1532:
1528:
1518:on 2022-10-11
1517:
1513:
1509:
1505:
1501:
1497:
1493:
1488:
1483:
1479:
1475:
1474:
1469:
1464:
1460:
1456:
1452:
1448:
1444:
1441:(in German).
1440:
1439:
1434:
1429:
1425:
1421:
1417:
1415:3-540-12177-3
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1354:
1349:
1345:
1341:
1337:
1333:
1331:0-8218-3227-1
1327:
1323:
1319:
1315:
1311:
1307:
1303:
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1262:
1257:
1253:
1249:
1248:
1243:
1239:
1235:
1231:
1227:
1225:3-540-13615-0
1221:
1217:
1213:
1209:
1205:
1201:
1197:
1193:
1192:
1187:
1180:
1175:
1172:
1168:
1163:
1160:
1156:
1151:
1148:
1144:
1140:
1135:
1132:
1128:
1123:
1120:
1116:
1111:
1108:
1104:
1099:
1096:
1092:
1087:
1084:
1080:
1075:
1072:
1068:
1063:
1060:
1056:
1051:
1048:
1044:
1040:
1035:
1032:
1028:
1023:
1020:
1016:
1012:
1007:
1004:
1000:
995:
992:
986:
984:
982:
978:
974:
970:
966:
962:
957:
953:
949:
944:
942:
936:
932:
908:
900:
896:
874:
866:
862:
858:
855:
849:
841:
837:
833:
830:
824:
821:
818:
808:
807:
806:
804:
800:
796:
788:
787:inner product
784:
780:
773:
772:tangent space
769:
764:
759:
755:
751:
747:
743:
739:
735:
731:
727:
723:
719:
715:
711:
707:
703:
699:
695:
691:
687:
683:
679:
675:
671:
663:
660:
658:
656:
652:
648:
644:
643:Stefan MĂŒller
640:
636:
632:
628:
624:
620:
616:
611:
600:
594:
586:
581:
578:
564:
562:
549:
542:
538:
534:
530:
522:
518:
517:David Hilbert
514:
509:
507:
501:
497:
490:
484:
474:
467:
463:
454:
450:
444:
440:
429:
425:
419:
414:
410:
406:
387:is less than
367:
359:
355:
344:
340:
325:
321:
310:
306:
294:
289:
286:
283:
279:
275:
269:
261:
258:
254:
246:
245:
244:
238:
232:
228:
205:
197:
191:
187:
183:
179:
173:
169:
154:
149:
147:
145:
143:
139:
135:
131:
127:
123:
119:
118:Maurice Janet
115:
111:
107:
103:
98:
96:
92:
88:
84:
80:
76:
71:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
2965:Moving frame
2960:Morse theory
2950:Gauge theory
2742:Tensor field
2671:Closed/Exact
2650:Vector field
2618:Distribution
2559:Hypercomplex
2554:Quaternionic
2291:Vector field
2249:Smooth atlas
2144:Applications
2072:Main results
1855:
1819:
1813:
1773:(1): 20â63.
1770:
1764:
1727:
1721:
1716:
1677:math/0402287
1667:
1661:
1625:
1619:
1614:
1585:
1579:
1574:
1539:
1520:. Retrieved
1516:the original
1477:
1471:
1442:
1436:
1393:
1357:
1351:
1305:
1251:
1245:
1203:
1174:
1162:
1150:
1134:
1122:
1110:
1098:
1086:
1074:
1062:
1050:
1043:Kuiper 1955b
1039:Kuiper 1955a
1034:
1022:
1006:
994:
947:
945:
934:
930:
902:
898:
894:
892:
802:
790:
782:
774:
762:
753:
749:
745:
737:
733:
729:
721:
717:
713:
709:
705:
701:
697:
693:
689:
685:
681:
677:
673:
669:
667:
661:
634:
612:
598:
592:
582:
576:
565:
531:of negative
510:
499:
495:
492:
472:
465:
461:
442:
438:
427:
423:
417:
416:
408:
404:
382:
230:
226:
189:
185:
177:
171:
167:
159:
152:
146:
121:
105:
101:
99:
94:
90:
78:
72:
35:
31:
29:
2910:Levi-Civita
2900:Generalized
2872:Connections
2822:Lie algebra
2754:Volume form
2655:Vector flow
2628:Pushforward
2623:Lie bracket
2522:Lie algebra
2487:G-structure
2276:Pushforward
2256:Submanifold
1628:: 683â689.
1588:: 545â556.
1155:Gromov 1986
1015:Gromov 1986
999:Taylor 2011
969:convolution
799:dot product
672:is a given
639:h-principle
126:h-principle
114:Ălie Cartan
3058:Categories
3033:Stratifold
2991:Diffeology
2787:Associated
2588:Symplectic
2573:Riemannian
2502:Hyperbolic
2429:Submersion
2337:HopfâRinow
2271:Submersion
2266:Smooth map
2104:Ricci flow
2053:Hyperbolic
1761:Nash, John
1712:Nash, John
1654:Ć verĂĄk, V.
1650:MĂŒller, S.
1522:2022-05-06
1487:1608.08301
1143:Isett 2018
1057:, Note 18.
768:linear map
758:derivative
720:+11)/2 if
700:+11)/2 if
50:into some
2915:Principal
2890:Ehresmann
2847:Subbundle
2837:Principal
2812:Fibration
2792:Cotangent
2664:Covectors
2517:Lie group
2497:Hermitian
2440:manifolds
2409:Immersion
2404:Foliation
2342:Noether's
2327:Frobenius
2322:De Rham's
2317:Darboux's
2208:Manifolds
2048:Hermitian
2001:Signature
1964:Sectional
1944:Curvature
1714:(1954). "
1512:119267892
1261:1202.1751
1179:Nash 1956
1127:Nash 1954
1027:Nash 1954
987:Citations
856:⋅
828:⟩
816:⟨
770:from the
742:embedding
621:, of the
561:ellipsoid
453:immersion
352:∂
345:α
337:∂
318:∂
311:α
303:∂
284:α
280:∑
160:Given an
89:of class
68:arclength
56:Isometric
3011:Orbifold
3006:K-theory
2996:Diffiety
2720:Pullback
2534:Oriented
2512:Kenmotsu
2492:Hadamard
2438:Types of
2387:Geodesic
2212:Glossary
2063:Kenmotsu
1976:Geodesic
1929:Glossary
1860:Springer
1854:(2011).
1810:Nash, J.
1704:55855605
1656:(2003).
1537:(1969).
1392:(1986).
1202:(1988).
943:(PDEs).
459:, where
196:pullback
83:analytic
48:embedded
2955:History
2938:Related
2852:Tangent
2830:)
2810:)
2777:Adjoint
2769:Bundles
2747:density
2645:Torsion
2611:Vectors
2603:Tensors
2586:)
2571:)
2567:,
2565:Pseudoâ
2544:Poisson
2477:Finsler
2472:Fibered
2467:Contact
2465:)
2457:Complex
2455:)
2424:Section
2130:Hilbert
2125:Finsler
1886:2744149
1844:0205266
1836:1970448
1795:0075639
1787:1969989
1752:0065993
1744:1969840
1696:1983780
1642:0075640
1602:0075640
1562:0238225
1504:3866888
1459:1037168
1424:0864505
1382:0283728
1374:1970760
1340:1909245
1294:2693636
1286:3090182
1266:Bibcode
1234:0936419
928:
916:
909:. When
402:
390:
224:
212:
2920:Vector
2905:Koszul
2885:Cartan
2880:Affine
2862:Vector
2857:Tensor
2842:Spinor
2832:Normal
2828:Stable
2782:Affine
2686:bundle
2638:bundle
2584:Almost
2507:KĂ€hler
2463:Almost
2453:Almost
2447:Closed
2347:Sard's
2303:(list)
2058:KĂ€hler
1954:Scalar
1949:tensor
1884:
1874:
1842:
1834:
1793:
1785:
1750:
1742:
1702:
1694:
1640:
1600:
1560:
1550:
1510:
1502:
1457:
1422:
1412:
1380:
1372:
1338:
1328:
1292:
1284:
1232:
1222:
756:, the
688:(with
680:, 3 â€
541:closed
527:. Any
481:which
432:be an
93:, 3 â€
87:smooth
3028:Sheaf
2802:Fiber
2578:Rizza
2549:Prime
2380:Local
2370:Curve
2232:Atlas
1959:Ricci
1832:JSTOR
1783:JSTOR
1740:JSTOR
1700:S2CID
1672:arXiv
1508:S2CID
1482:arXiv
1370:JSTOR
1290:S2CID
1256:arXiv
766:is a
716:+1)(3
449:short
176:, an
2895:Form
2797:Dual
2730:flow
2593:Tame
2569:Subâ
2482:Flat
2362:Maps
1872:ISBN
1548:ISBN
1410:ISBN
1326:ISBN
1220:ISBN
937:+ 1)
645:and
420:Let
411:+ 1)
239:for
233:+ 1)
128:and
116:and
100:The
60:path
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