17:
873:
865:
A non-holonomic solution to this relation would consist in the data of two functions, a differentiable function f(x), and a continuous function g(x), with g(x) nowhere vanishing. A holonomic solution gives rise to a non-holonomic solution by taking g(x) = f'(x). The space of non-holonomic solutions
780:
into a holonomic one in the class of non-holonomic solutions. Thus in the presence of h-principle, a differential topological problem reduces to an algebraic topological problem. More explicitly this means that apart from the topological obstruction there is no other obstruction to the existence of a
1141:
to holonomic ones but also can be arbitrarily well approximated by the holonomic ones (by going back and forth, like parallel parking in a limited space) – note that this approximates both the position and the angle of the car arbitrarily closely. This implies that, theoretically, it is possible to
788:
Many underdetermined partial differential equations satisfy the h-principle. However, the falsity of an h-principle is also an interesting statement, intuitively this means the objects being studied have non-trivial geometry that cannot be reduced to topology. As an example, embedded
958:
in complex numbers). Note that here there are no immersions of order 0, as those would need to turn back on themselves. Extending this to circles immersed in the plane – the immersion condition is precisely the condition that the derivative does not vanish – the
768:
In order to check whether a solution to our original equation exists, one can first check if there is a non-holonomic solution. Usually this is quite easy, and if there is no non-holonomic solution, then our original equation did not have any solutions.
621:
857:
A holonomic solution to this relation is a function whose derivative is nowhere vanishing, i.e. a strictly monotone differentiable function, either increasing or decreasing. The space of such functions consists of two disjoint
752:
518:
1335:-embedding (respectively, immersion). This is also a dense h-principle, and can be proven by an essentially similar "wrinkling" – or rather, circling – technique to the car in the plane, though it is much more involved.
986:
which requires the partial derivatives in each direction to not vanish and to be linearly independent, and the resulting analog of the Gauss map is a map to the
Stiefel manifold, or more generally between frame bundles.
1121:
253:
1536:
158:
390:
923:
1306:
848:
1689:
956:
1137:
A non-holonomic solution in this case, roughly speaking, corresponds to a motion of the car by sliding in the plane. In this case the non-holonomic solutions are not only
1244:
328:
288:
1477:
1441:
1053:
1876:
1840:
1774:
1747:
1716:
1660:
1633:
1599:
1572:
1408:
1333:
1271:
1198:
1167:
1497:
1033:
1013:
529:
1887:
David Spring, Convex integration theory - solutions to the h-principle in geometry and topology, Monographs in
Mathematics 92, Birkhauser-Verlag, 1998
632:
398:
995:
As another simple example, consider a car moving in the plane. The position of a car in the plane is determined by three parameters: two coordinates
2017:
982:
are much further-reaching generalizations, and much more involved, but similar in principle – immersion requires the derivative to have rank
888:
This trivial example has nontrivial generalizations: extending this to immersions of a circle into itself classifies them by order (or
1957:
1935:
1061:
2022:
166:
1945:
60:
960:
877:
68:
1142:
parallel park in any space longer than the length of your car. It also implies that, in a contact 3 manifold, any curve is
41:
1718:
because a small oscillating sphere would provide a large lower bound for the principal curvatures, and therefore for the
45:
16:
1502:
978:, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes of maps of
896:
and applying the above analysis to the resulting monotone map – the linear map corresponds to multiplying angle:
98:
893:
793:
in a symplectic manifold do not satisfy an h-principle, to prove this one needs to find invariants coming from
333:
1212:
899:
794:
52:
PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, and other areas.
1208:
1131:
790:
49:
1276:
63:
and
Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to
1665:
1035:
for the location (a good choice is the location of the midpoint between the back wheels) and an angle
1806:
S. Smale, The classification of immersions of spheres in
Euclidean spaces. Ann. of Math(2) 69 (1959)
928:
1605:
785:
is much easier to handle and can be addressed with the obstruction theory for topological bundles.
813:
1984:
1222:
1541:
Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.
1359:
Here we list a few counter-intuitive results which can be proved by applying the h-principle:
301:
261:
1953:
1931:
1777:
1446:
872:
1413:
1038:
1994:
975:
56:
1854:
1818:
1752:
1725:
1694:
1638:
1611:
1577:
1550:
1386:
1311:
1249:
1176:
1145:
1719:
1544:
1170:
21:
1055:
which describes the orientation of the car. The motion of the car satisfies the equation
869:
Thus the inclusion of holonomic into non-holonomic solutions satisfies the h-principle.
866:
again consists of two disjoint convex sets, according as g(x) is positive or negative.
616:{\displaystyle y_{j}={\partial ^{k}f \over \partial u_{j_{1}}\ldots \partial u_{j_{k}}}.}
971:
and showing that this satisfies an h-principle; here again order 0 is more complicated.
862:: the increasing ones and the decreasing ones, and has the homotopy type of two points.
810:
Perhaps the simplest partial differential relation is for the derivative to not vanish:
1482:
1018:
998:
964:
889:
881:
880:
shows that immersions of the circle in the plane satisfy an h-principle, expressed by
761:, and a solution of the system which is also solution of our original PDE is called a
2011:
1173:
curve. This last property is stronger than the general h-principle; it is called the
1999:
1968:
979:
1921:
1913:
67:, particularly for immersions. The first evidence of h-principle appeared in the
747:{\displaystyle \Psi _{}^{}(u_{1},u_{2},\dots ,u_{m},y_{1},y_{2},\dots ,y_{N})=0}
513:{\displaystyle \Psi _{}^{}(u_{1},u_{2},\dots ,u_{m},y_{1},y_{2},\dots ,y_{N})=0}
29:
859:
1216:
1138:
968:
1749:
this has to be equal to 1 everywhere, the Gauss curvature of the standard
974:
Smale's classification of immersions of spheres as the homotopy groups of
1878:
Isometric
Imbeddings I, II. Nederl. Acad. Wetensch. Proc. Ser A 58 (1955)
1127:
777:
64:
1797:
M. W. Hirsch, Immersions of manifold. Trans. Amer. Math. Soc. 93 (1959)
1126:
since a non-skidding car must move in the direction of its wheels. In
1344:
Removal of
Singularities technique developed by Gromov and Eliashberg
20:
The homotopy principle generalizes such results as Smale's proof of
1989:
1722:
of the immersed sphere, but on the other hand if the immersion is
871:
15:
1383:|. Then there is a continuous one-parameter family of functions
1308:
or larger can be arbitrarily well approximated by an isometric
854:
differential relation, as this is a function in one variable.
1116:{\displaystyle {\dot {x}}\sin \alpha ={\dot {y}}\cos \alpha .}
1898:
Mathematical
Omnibus: Thirty Lectures on Classic Mathematics
248:{\displaystyle \Psi (u_{1},u_{2},\dots ,u_{m},J_{f}^{k})=0}
91:
which satisfies a partial differential equation of degree
75:
embedding theorem and the Smale–Hirsch immersion theorem.
781:
holonomic solution. The topological problem of finding a
392:
Then our original equation can be thought as a system of
1350:
Convex integration based on the work of Nash and Kuiper.
1347:
Sheaf technique based on the work of Smale and Hirsch.
1857:
1821:
1755:
1728:
1697:
1668:
1641:
1614:
1580:
1553:
1505:
1485:
1449:
1416:
1389:
1314:
1279:
1252:
1225:
1179:
1148:
1064:
1041:
1021:
1001:
931:
902:
816:
635:
532:
401:
336:
304:
264:
169:
101:
1967:
De Lellis, Camillo; Székelyhidi, László Jr. (2012).
523:and some number of equations of the following type
1870:
1834:
1768:
1741:
1710:
1683:
1654:
1627:
1593:
1566:
1530:
1491:
1471:
1435:
1402:
1327:
1300:
1265:
1238:
1192:
1161:
1115:
1047:
1027:
1007:
950:
917:
842:
746:
615:
512:
384:
322:
282:
247:
152:
1920:Eliashberg, Y.; Mishachev, N.; Ariki, S. (2002).
71:. This was followed by the Nash–Kuiper isometric
1973:-principle and the equations of fluid dynamics"
1842:Isometric Imbedding. Ann. of Math(2) 60 (1954)
1547:without creasing or tearing can be done using
1207:While this example is simple, compare to the
8:
1531:{\displaystyle \operatorname {grad} (f_{t})}
1130:terms, not all paths in the task space are
153:{\displaystyle (u_{1},u_{2},\dots ,u_{m})}
1998:
1988:
1862:
1856:
1826:
1820:
1760:
1754:
1733:
1727:
1702:
1696:
1675:
1671:
1670:
1667:
1646:
1640:
1619:
1613:
1606:Nash-Kuiper C isometric embedding theorem
1585:
1579:
1558:
1552:
1519:
1504:
1484:
1454:
1448:
1421:
1415:
1394:
1388:
1319:
1313:
1286:
1281:
1278:
1257:
1251:
1230:
1224:
1184:
1178:
1153:
1147:
1090:
1089:
1066:
1065:
1063:
1040:
1020:
1000:
967:by considering the homotopy class of the
942:
930:
901:
815:
729:
710:
697:
684:
665:
652:
642:
640:
634:
599:
594:
576:
571:
553:
546:
537:
531:
495:
476:
463:
450:
431:
418:
408:
406:
400:
385:{\displaystyle y_{1},y_{2},\dots ,y_{N}.}
373:
354:
341:
335:
314:
309:
303:
274:
269:
263:
230:
225:
212:
193:
180:
168:
141:
122:
109:
100:
1608:, in particular implies that there is a
1790:
918:{\displaystyle \theta \mapsto n\theta }
290:stands for all partial derivatives of
776:if any non-holonomic solution can be
48:(PDRs). The h-principle is good for
7:
298:. Let us exchange every variable in
1662:into an arbitrarily small ball of
1363:Cone eversion. Consider functions
1301:{\displaystyle \mathbf {R} ^{m+1}}
1231:
637:
587:
564:
550:
403:
170:
83:Assume we want to find a function
14:
1930:. American Mathematical Society.
1635:isometric immersion of the round
40:) is a very general way to solve
1684:{\displaystyle \mathbb {R} ^{3}}
1282:
2000:10.1090/S0273-0979-2012-01376-9
2018:Partial differential equations
1950:Partial differential relations
1525:
1512:
951:{\displaystyle z\mapsto z^{n}}
935:
906:
831:
825:
735:
645:
501:
411:
330:for new independent variables
236:
173:
147:
102:
46:partial differential relations
42:partial differential equations
1:
1339:Ways to prove the h-principle
892:), by lifting the map to the
1246:) embedding or immersion of
843:{\displaystyle f'(x)\neq 0.}
1691:. This immersion cannot be
1239:{\displaystyle C^{\infty }}
44:(PDEs), and more generally
2039:
1896:D. Fuchs, S. Tabachnikov,
55:The theory was started by
1916:, translation Kiki Hudson
1914:Embeddings and immersions
1538:is not zero at any point.
961:Whitney–Graustein theorem
878:Whitney–Graustein theorem
795:pseudo-holomorphic curves
774:satisfies the h-principle
323:{\displaystyle J_{f}^{k}}
283:{\displaystyle J_{f}^{k}}
69:Whitney–Graustein theorem
1472:{\displaystyle f_{1}=-f}
894:universal covering space
160:. One can rewrite it as
2023:Mathematical principles
1436:{\displaystyle f_{0}=f}
1048:{\displaystyle \alpha }
1872:
1836:
1770:
1743:
1712:
1685:
1656:
1629:
1595:
1568:
1532:
1493:
1473:
1437:
1404:
1329:
1302:
1267:
1240:
1215:, which says that any
1209:Nash embedding theorem
1194:
1163:
1117:
1049:
1029:
1009:
952:
919:
885:
844:
783:non-holonomic solution
759:non-holonomic solution
748:
617:
514:
386:
324:
284:
249:
154:
25:
1977:Bull. Amer. Math. Soc
1873:
1871:{\displaystyle C^{1}}
1837:
1835:{\displaystyle C^{1}}
1771:
1769:{\displaystyle S^{2}}
1744:
1742:{\displaystyle C^{2}}
1713:
1711:{\displaystyle C^{2}}
1686:
1657:
1655:{\displaystyle S^{2}}
1630:
1628:{\displaystyle C^{1}}
1596:
1594:{\displaystyle S^{2}}
1569:
1567:{\displaystyle C^{1}}
1533:
1494:
1474:
1438:
1405:
1403:{\displaystyle f_{t}}
1330:
1328:{\displaystyle C^{1}}
1303:
1268:
1266:{\displaystyle M^{m}}
1241:
1195:
1193:{\displaystyle C^{0}}
1164:
1162:{\displaystyle C^{0}}
1118:
1050:
1030:
1010:
953:
920:
875:
850:Properly, this is an
845:
749:
618:
515:
387:
325:
285:
250:
155:
19:
1923:Introduction to the
1855:
1819:
1753:
1726:
1695:
1666:
1639:
1612:
1578:
1551:
1503:
1483:
1447:
1414:
1387:
1312:
1277:
1250:
1223:
1177:
1146:
1062:
1039:
1019:
999:
963:classified these by
929:
900:
814:
633:
530:
399:
334:
302:
262:
167:
99:
1213:Nash–Kuiper theorem
1211:, specifically the
644:
410:
319:
279:
235:
1868:
1832:
1766:
1739:
1708:
1681:
1652:
1625:
1591:
1564:
1528:
1489:
1469:
1433:
1400:
1325:
1298:
1263:
1236:
1190:
1159:
1113:
1045:
1025:
1005:
991:A car in the plane
948:
915:
886:
840:
806:Monotone functions
763:holonomic solution
744:
636:
613:
510:
402:
382:
320:
305:
280:
265:
245:
221:
150:
95:, in co-ordinates
34:homotopy principle
26:
1912:Masahisa Adachi,
1778:Theorema Egregium
1492:{\displaystyle t}
1202:dense h-principle
1098:
1074:
1028:{\displaystyle y}
1008:{\displaystyle x}
976:Stiefel manifolds
608:
294:up to order
2030:
2004:
2002:
1992:
1963:
1941:
1900:
1894:
1888:
1885:
1879:
1877:
1875:
1874:
1869:
1867:
1866:
1849:
1843:
1841:
1839:
1838:
1833:
1831:
1830:
1813:
1807:
1804:
1798:
1795:
1775:
1773:
1772:
1767:
1765:
1764:
1748:
1746:
1745:
1740:
1738:
1737:
1717:
1715:
1714:
1709:
1707:
1706:
1690:
1688:
1687:
1682:
1680:
1679:
1674:
1661:
1659:
1658:
1653:
1651:
1650:
1634:
1632:
1631:
1626:
1624:
1623:
1600:
1598:
1597:
1592:
1590:
1589:
1573:
1571:
1570:
1565:
1563:
1562:
1537:
1535:
1534:
1529:
1524:
1523:
1498:
1496:
1495:
1490:
1478:
1476:
1475:
1470:
1459:
1458:
1442:
1440:
1439:
1434:
1426:
1425:
1409:
1407:
1406:
1401:
1399:
1398:
1334:
1332:
1331:
1326:
1324:
1323:
1307:
1305:
1304:
1299:
1297:
1296:
1285:
1272:
1270:
1269:
1264:
1262:
1261:
1245:
1243:
1242:
1237:
1235:
1234:
1199:
1197:
1196:
1191:
1189:
1188:
1168:
1166:
1165:
1160:
1158:
1157:
1122:
1120:
1119:
1114:
1100:
1099:
1091:
1076:
1075:
1067:
1054:
1052:
1051:
1046:
1034:
1032:
1031:
1026:
1014:
1012:
1011:
1006:
957:
955:
954:
949:
947:
946:
924:
922:
921:
916:
849:
847:
846:
841:
824:
753:
751:
750:
745:
734:
733:
715:
714:
702:
701:
689:
688:
670:
669:
657:
656:
643:
641:
622:
620:
619:
614:
609:
607:
606:
605:
604:
603:
583:
582:
581:
580:
562:
558:
557:
547:
542:
541:
519:
517:
516:
511:
500:
499:
481:
480:
468:
467:
455:
454:
436:
435:
423:
422:
409:
407:
391:
389:
388:
383:
378:
377:
359:
358:
346:
345:
329:
327:
326:
321:
318:
313:
289:
287:
286:
281:
278:
273:
254:
252:
251:
246:
234:
229:
217:
216:
198:
197:
185:
184:
159:
157:
156:
151:
146:
145:
127:
126:
114:
113:
57:Yakov Eliashberg
2038:
2037:
2033:
2032:
2031:
2029:
2028:
2027:
2008:
2007:
1966:
1960:
1944:
1938:
1919:
1909:
1907:Further reading
1904:
1903:
1895:
1891:
1886:
1882:
1858:
1853:
1852:
1850:
1846:
1822:
1817:
1816:
1814:
1810:
1805:
1801:
1796:
1792:
1787:
1756:
1751:
1750:
1729:
1724:
1723:
1720:Gauss curvature
1698:
1693:
1692:
1669:
1664:
1663:
1642:
1637:
1636:
1615:
1610:
1609:
1581:
1576:
1575:
1554:
1549:
1548:
1545:Sphere eversion
1515:
1501:
1500:
1481:
1480:
1450:
1445:
1444:
1417:
1412:
1411:
1390:
1385:
1384:
1379:) = |
1371:without origin
1357:
1341:
1315:
1310:
1309:
1280:
1275:
1274:
1253:
1248:
1247:
1226:
1221:
1220:
1180:
1175:
1174:
1149:
1144:
1143:
1060:
1059:
1037:
1036:
1017:
1016:
997:
996:
993:
938:
927:
926:
898:
897:
817:
812:
811:
808:
803:
801:Simple examples
725:
706:
693:
680:
661:
648:
631:
630:
595:
590:
572:
567:
563:
549:
548:
533:
528:
527:
491:
472:
459:
446:
427:
414:
397:
396:
369:
350:
337:
332:
331:
300:
299:
260:
259:
208:
189:
176:
165:
164:
137:
118:
105:
97:
96:
81:
50:underdetermined
22:sphere eversion
12:
11:
5:
2036:
2034:
2026:
2025:
2020:
2010:
2009:
2006:
2005:
1964:
1958:
1942:
1936:
1917:
1908:
1905:
1902:
1901:
1889:
1880:
1865:
1861:
1851:N. Kuiper, On
1844:
1829:
1825:
1808:
1799:
1789:
1788:
1786:
1783:
1782:
1781:
1763:
1759:
1736:
1732:
1705:
1701:
1678:
1673:
1649:
1645:
1622:
1618:
1602:
1588:
1584:
1574:immersions of
1561:
1557:
1542:
1539:
1527:
1522:
1518:
1514:
1511:
1508:
1488:
1468:
1465:
1462:
1457:
1453:
1432:
1429:
1424:
1420:
1397:
1393:
1356:
1355:Some paradoxes
1353:
1352:
1351:
1348:
1345:
1340:
1337:
1322:
1318:
1295:
1292:
1289:
1284:
1260:
1256:
1233:
1229:
1187:
1183:
1156:
1152:
1124:
1123:
1112:
1109:
1106:
1103:
1097:
1094:
1088:
1085:
1082:
1079:
1073:
1070:
1044:
1024:
1004:
992:
989:
965:turning number
945:
941:
937:
934:
914:
911:
908:
905:
890:winding number
882:turning number
839:
836:
833:
830:
827:
823:
820:
807:
804:
802:
799:
755:
754:
743:
740:
737:
732:
728:
724:
721:
718:
713:
709:
705:
700:
696:
692:
687:
683:
679:
676:
673:
668:
664:
660:
655:
651:
647:
639:
626:A solution of
624:
623:
612:
602:
598:
593:
589:
586:
579:
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