98:
2715:
1208:
2413:
2488:
1026:
1050:
1996:
2237:
1458:
682:
2710:{\displaystyle \langle R_{N}(u,v)w,z\rangle =\langle R_{M}(u,v)w,z\rangle +\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle \,.}
284:
1664:
1538:
913:
1203:{\displaystyle L=-\mathbf {r} _{u}\cdot \mathbf {H} \cdot \mathbf {r} _{u}\,,\quad M=-\mathbf {r} _{u}\cdot \mathbf {H} \cdot \mathbf {r} _{v}\,,\quad N=-\mathbf {r} _{v}\cdot \mathbf {H} \cdot \mathbf {r} _{v}\,.}
1756:
1859:
866:
781:
2135:
2408:{\displaystyle \langle R(u,v)w,z\rangle =\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle .}
1362:
586:
386:
2181:
71:
2214:
1806:
1830:
1779:
199:
3074:
1594:
1469:
1021:{\displaystyle L=\mathbf {r} _{uu}\cdot \mathbf {n} \,,\quad M=\mathbf {r} _{uv}\cdot \mathbf {n} \,,\quad N=\mathbf {r} _{vv}\cdot \mathbf {n} \,.}
1991:{\displaystyle \mathrm {I} \!\mathrm {I} (v,w)=\langle S(v),w\rangle n=-\langle \nabla _{v}n,w\rangle n=\langle n,\nabla _{v}w\rangle n\,,}
1684:
2860:
814:
693:
2907:
2995:
2932:
2811:
2792:
2773:
2066:
40:
2040:(which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of
3084:
2882:
1833:
3069:
2832:
1453:{\displaystyle \mathbf {n} ={\frac {\mathbf {r} _{1}\times \mathbf {r} _{2}}{|\mathbf {r} _{1}\times \mathbf {r} _{2}|}}\,.}
677:{\displaystyle \mathbf {n} ={\frac {\mathbf {r} _{u}\times \mathbf {r} _{v}}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}\,.}
3079:
2962:
2736:
310:
130:
2902:
2420:
1809:
2838:
3025:
3000:
2922:
2224:
1032:
2853:
2143:
1843:
More generally, on a
Riemannian manifold, the second fundamental form is an equivalent way to describe the
1350:
is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors
574:
is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors
2977:
2967:
2746:
2726:
1259:
462:
74:
2874:
2751:
20:
2828:
2184:
2014:
126:
50:
3015:
2987:
2942:
2438:
86:
78:
2189:
2947:
2897:
2846:
2731:
170:
111:
2807:
2788:
2769:
2425:
2024:
1544:
3089:
3010:
2912:
279:{\displaystyle z=L{\frac {x^{2}}{2}}+Mxy+N{\frac {y^{2}}{2}}+{\text{higher order terms}}\,,}
186:
1788:
3005:
2917:
2220:
1815:
1764:
1675:
44:
97:
3038:
3033:
2952:
2741:
2028:
1844:
1036:
430:
The second fundamental form of a general parametric surface is defined as follows. Let
302:
32:
3063:
2957:
2829:
Geometry of the Second
Fundamental Form: Curvature Properties and Variational Aspects
2057:
2041:
1837:
1332:
552:
36:
2431:
For general
Riemannian manifolds one has to add the curvature of ambient space; if
1659:{\displaystyle b_{\alpha \beta }=r_{,\alpha \beta }^{\ \ \,\gamma }n_{\gamma }\,.}
3043:
2228:
2053:
904:
82:
1533:{\displaystyle \mathrm {I\!I} =b_{\alpha \beta }\,du^{\alpha }\,du^{\beta }\,.}
2056:. In that case it is a quadratic form on the tangent space with values in the
2034:
The sign of the second fundamental form depends on the choice of direction of
2892:
2870:
1782:
2467:
with induced metric can be expressed using the second fundamental form and
3048:
1570:-plane are given by the projections of the second partial derivatives of
891:-plane are given by the projections of the second partial derivatives of
81:. More generally, such a quadratic form is defined for a smooth immersed
1044:, the second fundamental form coefficients can be computed as follows:
160:
1751:{\displaystyle \mathrm {I\!I} (v,w)=-\langle d\nu (v),w\rangle \nu }
1262:
of two variables. It is common to denote the partial derivatives of
465:
of two variables. It is common to denote the partial derivatives of
96:
861:{\displaystyle {\begin{bmatrix}L&M\\M&N\end{bmatrix}}\,.}
776:{\displaystyle \mathrm {I\!I} =L\,du^{2}+2M\,du\,dv+N\,dv^{2}\,,}
289:
and the second fundamental form at the origin in the coordinates
2804:
A Comprehensive introduction to differential geometry (Volume 3)
2842:
77:, it serves to define extrinsic invariants of the surface, its
2130:{\displaystyle \mathrm {I\!I} (v,w)=(\nabla _{v}w)^{\bot }\,,}
2052:
The second fundamental form can be generalized to arbitrary
1218:
The second fundamental form of a general parametric surface
422:, and define the second fundamental form in the same way.
403:, one can choose the coordinate system so that the plane
129:. First suppose that the surface is the graph of a twice
2023:
a field of normal vectors on the hypersurface. (If the
2764:
Guggenheimer, Heinrich (1977). "Chapter 10. Surfaces".
823:
2491:
2424:, as it may be viewed as a generalization of Gauss's
2240:
2192:
2146:
2069:
1862:
1818:
1791:
1767:
1687:
1597:
1472:
1365:
1053:
916:
817:
696:
589:
313:
202:
53:
3024:
2986:
2931:
2881:
2783:Kobayashi, Shoshichi & Nomizu, Katsumi (1996).
2709:
2407:
2208:
2175:
2129:
2031:, then the second fundamental form is symmetric.)
1990:
1824:
1800:
1773:
1750:
1658:
1582:and can be computed in terms of the normal vector
1532:
1463:The second fundamental form is usually written as
1452:
1202:
1020:
860:
775:
687:The second fundamental form is usually written as
676:
380:
278:
65:
2679:
2650:
2615:
2586:
2378:
2349:
2314:
2285:
2074:
1868:
1692:
1477:
701:
381:{\displaystyle L\,dx^{2}+2M\,dx\,dy+N\,dy^{2}\,.}
58:
16:Quadratic form related to curvatures of surfaces
1289:. Regularity of the parametrization means that
505:. Regularity of the parametrization means that
2854:
1246:be a regular parametrization of a surface in
449:be a regular parametrization of a surface in
8:
2785:Foundations of Differential Geometry, Vol. 2
2700:
2642:
2636:
2578:
2572:
2535:
2529:
2492:
2399:
2341:
2335:
2277:
2271:
2241:
1978:
1956:
1947:
1925:
1913:
1892:
1742:
1718:
2231:can be described by the following formula:
2861:
2847:
2839:
1678:, the second fundamental form is given by
2703:
2680:
2674:
2651:
2645:
2616:
2610:
2587:
2581:
2542:
2499:
2490:
2379:
2373:
2350:
2344:
2315:
2309:
2286:
2280:
2239:
2197:
2191:
2167:
2154:
2145:
2123:
2117:
2104:
2070:
2068:
1984:
1969:
1932:
1869:
1863:
1861:
1817:
1790:
1766:
1688:
1686:
1652:
1646:
1636:
1629:
1618:
1602:
1596:
1526:
1520:
1512:
1506:
1498:
1489:
1473:
1471:
1446:
1438:
1432:
1427:
1417:
1412:
1406:
1398:
1393:
1383:
1378:
1374:
1366:
1364:
1196:
1190:
1185:
1176:
1167:
1162:
1147:
1141:
1136:
1127:
1118:
1113:
1098:
1092:
1087:
1078:
1069:
1064:
1052:
1014:
1009:
997:
992:
980:
975:
963:
958:
946:
941:
929:
924:
915:
854:
818:
816:
769:
763:
755:
742:
735:
720:
712:
697:
695:
670:
662:
656:
651:
641:
636:
630:
622:
617:
607:
602:
598:
590:
588:
374:
368:
360:
347:
340:
325:
317:
312:
272:
267:
253:
247:
218:
212:
201:
54:
52:
903:and can be computed with the aid of the
2048:Generalization to arbitrary codimension
2176:{\displaystyle (\nabla _{v}w)^{\bot }}
1576:at that point onto the normal line to
1325:, and hence span the tangent plane to
897:at that point onto the normal line to
545:, and hence span the tangent plane to
193:at (0,0) starts with quadratic terms:
2183:denotes the orthogonal projection of
1670:Hypersurface in a Riemannian manifold
101:Definition of second fundamental form
7:
2787:(New ed.). Wiley-Interscience.
1561:at a given point in the parametric
885:at a given point in the parametric
163:to the surface at the origin. Then
2908:Radius of curvature (applications)
2681:
2675:
2652:
2646:
2617:
2611:
2588:
2582:
2380:
2374:
2351:
2345:
2316:
2310:
2287:
2281:
2194:
2168:
2151:
2118:
2101:
2075:
2071:
1966:
1929:
1870:
1864:
1693:
1689:
1478:
1474:
702:
698:
59:
55:
14:
3075:Differential geometry of surfaces
2996:Curvature of Riemannian manifolds
1331:at each point. Equivalently, the
1307:are linearly independent for any
551:at each point. Equivalently, the
527:are linearly independent for any
110:The second fundamental form of a
1428:
1413:
1394:
1379:
1367:
1186:
1177:
1163:
1137:
1128:
1114:
1088:
1079:
1065:
1010:
993:
976:
959:
942:
925:
652:
637:
618:
603:
591:
185:vanish at (0,0). Therefore, the
73:(read "two"). Together with the
1834:vector-valued differential form
1151:
1102:
984:
950:
66:{\displaystyle \mathrm {I\!I} }
2833:Katholieke Universiteit Leuven
2697:
2685:
2668:
2656:
2633:
2621:
2604:
2592:
2560:
2548:
2517:
2505:
2396:
2384:
2367:
2355:
2332:
2320:
2303:
2291:
2259:
2247:
2164:
2147:
2114:
2097:
2091:
2079:
1904:
1898:
1886:
1874:
1836:, and the brackets denote the
1733:
1727:
1709:
1697:
1439:
1407:
663:
631:
125:was introduced and studied by
1:
1545:Einstein summation convention
2437:is a manifold embedded in a
2209:{\displaystyle \nabla _{v}w}
2017:of the ambient manifold and
1543:The equation above uses the
131:continuously differentiable
3106:
2476:, the curvature tensor of
2452:then the curvature tensor
2060:and it can be defined by
43:in the three-dimensional
3026:Curvature of connections
3001:Riemann curvature tensor
2923:Total absolute curvature
2802:Spivak, Michael (1999).
2216:onto the normal bundle.
1213:
808:of the tangent plane is
786:its matrix in the basis
3085:Curvature (mathematics)
2973:Second fundamental form
2963:GaussâCodazzi equations
2827:Steven Verpoort (2008)
2737:GaussâCodazzi equations
1224:is defined as follows.
25:second fundamental form
2978:Third fundamental form
2968:First fundamental form
2933:Differential geometry
2903:FrenetâSerret formulas
2883:Differential geometry
2747:Third fundamental form
2727:First fundamental form
2711:
2409:
2210:
2177:
2131:
1992:
1826:
1802:
1775:
1752:
1660:
1534:
1454:
1260:vector-valued function
1204:
1022:
862:
777:
678:
463:vector-valued function
382:
280:
102:
75:first fundamental form
67:
3070:Differential geometry
2875:differential geometry
2806:. Publish or Perish.
2766:Differential Geometry
2752:Tautological one-form
2712:
2410:
2211:
2178:
2132:
1993:
1853:) of a hypersurface,
1827:
1803:
1801:{\displaystyle d\nu }
1776:
1753:
1661:
1535:
1455:
1205:
1033:signed distance field
1023:
863:
778:
679:
383:
281:
152:, and that the plane
100:
68:
47:, usually denoted by
21:differential geometry
2943:Principal curvatures
2489:
2238:
2190:
2185:covariant derivative
2144:
2067:
2015:covariant derivative
1860:
1840:of Euclidean space.
1825:{\displaystyle \nu }
1816:
1789:
1774:{\displaystyle \nu }
1765:
1685:
1595:
1470:
1363:
1214:Physicist's notation
1051:
914:
815:
694:
587:
311:
200:
79:principal curvatures
51:
3080:Riemannian geometry
3016:Sectional curvature
2988:Riemannian geometry
2869:Various notions of
2439:Riemannian manifold
2418:This is called the
1641:
391:For a smooth point
171:partial derivatives
87:Riemannian manifold
2948:Gaussian curvature
2898:Torsion of a curve
2732:Gaussian curvature
2707:
2405:
2206:
2173:
2127:
1988:
1822:
1798:
1771:
1748:
1656:
1614:
1530:
1450:
1200:
1018:
858:
848:
773:
674:
426:Classical notation
378:
276:
269:higher order terms
112:parametric surface
103:
63:
3057:
3056:
2426:Theorema Egregium
2044:of the surface).
2025:affine connection
1635:
1632:
1550:The coefficients
1444:
1319:in the domain of
871:The coefficients
668:
539:in the domain of
270:
262:
227:
3097:
3011:Scalar curvature
2913:Affine curvature
2863:
2856:
2849:
2840:
2817:
2798:
2779:
2716:
2714:
2713:
2708:
2684:
2678:
2655:
2649:
2620:
2614:
2591:
2585:
2547:
2546:
2504:
2503:
2481:
2475:
2466:
2460:
2451:
2436:
2414:
2412:
2411:
2406:
2383:
2377:
2354:
2348:
2319:
2313:
2290:
2284:
2225:curvature tensor
2215:
2213:
2212:
2207:
2202:
2201:
2182:
2180:
2179:
2174:
2172:
2171:
2159:
2158:
2136:
2134:
2133:
2128:
2122:
2121:
2109:
2108:
2078:
2039:
2022:
2012:
1997:
1995:
1994:
1989:
1974:
1973:
1937:
1936:
1873:
1867:
1852:
1831:
1829:
1828:
1823:
1807:
1805:
1804:
1799:
1780:
1778:
1777:
1772:
1757:
1755:
1754:
1749:
1696:
1665:
1663:
1662:
1657:
1651:
1650:
1640:
1633:
1630:
1628:
1610:
1609:
1587:
1581:
1575:
1569:
1560:
1539:
1537:
1536:
1531:
1525:
1524:
1511:
1510:
1497:
1496:
1481:
1459:
1457:
1456:
1451:
1445:
1443:
1442:
1437:
1436:
1431:
1422:
1421:
1416:
1410:
1404:
1403:
1402:
1397:
1388:
1387:
1382:
1375:
1370:
1355:
1349:
1330:
1324:
1318:
1306:
1297:
1288:
1284:
1273:
1268:with respect to
1267:
1257:
1251:
1245:
1223:
1209:
1207:
1206:
1201:
1195:
1194:
1189:
1180:
1172:
1171:
1166:
1146:
1145:
1140:
1131:
1123:
1122:
1117:
1097:
1096:
1091:
1082:
1074:
1073:
1068:
1043:
1027:
1025:
1024:
1019:
1013:
1005:
1004:
996:
979:
971:
970:
962:
945:
937:
936:
928:
902:
896:
890:
884:
867:
865:
864:
859:
853:
852:
807:
782:
780:
779:
774:
768:
767:
725:
724:
705:
683:
681:
680:
675:
669:
667:
666:
661:
660:
655:
646:
645:
640:
634:
628:
627:
626:
621:
612:
611:
606:
599:
594:
579:
573:
550:
544:
538:
526:
515:
504:
493:
482:
476:
471:with respect to
470:
460:
454:
448:
421:
415:
409:
402:
396:
387:
385:
384:
379:
373:
372:
330:
329:
300:
285:
283:
282:
277:
271:
268:
263:
258:
257:
248:
228:
223:
222:
213:
187:Taylor expansion
184:
178:
173:with respect to
168:
158:
151:
124:
118:
72:
70:
69:
64:
62:
3105:
3104:
3100:
3099:
3098:
3096:
3095:
3094:
3060:
3059:
3058:
3053:
3020:
3006:Ricci curvature
2982:
2934:
2927:
2918:Total curvature
2884:
2877:
2867:
2824:
2814:
2801:
2795:
2782:
2776:
2763:
2760:
2723:
2538:
2495:
2487:
2486:
2477:
2473:
2468:
2462:
2458:
2453:
2441:
2432:
2236:
2235:
2221:Euclidean space
2193:
2188:
2187:
2163:
2150:
2142:
2141:
2113:
2100:
2065:
2064:
2050:
2035:
2018:
2008:
2002:
1965:
1928:
1858:
1857:
1848:
1814:
1813:
1787:
1786:
1763:
1762:
1683:
1682:
1676:Euclidean space
1672:
1642:
1598:
1593:
1592:
1583:
1577:
1571:
1562:
1559:
1551:
1516:
1502:
1485:
1468:
1467:
1426:
1411:
1405:
1392:
1377:
1376:
1361:
1360:
1351:
1348:
1341:
1335:
1326:
1320:
1308:
1305:
1299:
1296:
1290:
1286:
1283:
1275:
1269:
1263:
1253:
1247:
1228:
1219:
1216:
1184:
1161:
1135:
1112:
1086:
1063:
1049:
1048:
1039:
991:
957:
923:
912:
911:
898:
892:
886:
872:
847:
846:
841:
835:
834:
829:
819:
813:
812:
805:
796:
787:
759:
716:
692:
691:
650:
635:
629:
616:
601:
600:
585:
584:
575:
572:
563:
555:
546:
540:
528:
525:
517:
514:
506:
503:
495:
492:
484:
478:
472:
466:
456:
450:
431:
428:
417:
411:
404:
398:
392:
364:
321:
309:
308:
290:
249:
214:
198:
197:
180:
174:
164:
153:
134:
120:
114:
108:
95:
49:
48:
45:Euclidean space
17:
12:
11:
5:
3103:
3101:
3093:
3092:
3087:
3082:
3077:
3072:
3062:
3061:
3055:
3054:
3052:
3051:
3046:
3041:
3039:Torsion tensor
3036:
3034:Curvature form
3030:
3028:
3022:
3021:
3019:
3018:
3013:
3008:
3003:
2998:
2992:
2990:
2984:
2983:
2981:
2980:
2975:
2970:
2965:
2960:
2955:
2953:Mean curvature
2950:
2945:
2939:
2937:
2929:
2928:
2926:
2925:
2920:
2915:
2910:
2905:
2900:
2895:
2889:
2887:
2879:
2878:
2868:
2866:
2865:
2858:
2851:
2843:
2837:
2836:
2823:
2822:External links
2820:
2819:
2818:
2812:
2799:
2793:
2780:
2774:
2759:
2756:
2755:
2754:
2749:
2744:
2742:Shape operator
2739:
2734:
2729:
2722:
2719:
2718:
2717:
2706:
2702:
2699:
2696:
2693:
2690:
2687:
2683:
2677:
2673:
2670:
2667:
2664:
2661:
2658:
2654:
2648:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2619:
2613:
2609:
2606:
2603:
2600:
2597:
2594:
2590:
2584:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2545:
2541:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2502:
2498:
2494:
2471:
2456:
2421:Gauss equation
2416:
2415:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2382:
2376:
2372:
2369:
2366:
2363:
2360:
2357:
2353:
2347:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2318:
2312:
2308:
2305:
2302:
2299:
2296:
2293:
2289:
2283:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2205:
2200:
2196:
2170:
2166:
2162:
2157:
2153:
2149:
2138:
2137:
2126:
2120:
2116:
2112:
2107:
2103:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2077:
2073:
2049:
2046:
2004:
1999:
1998:
1987:
1983:
1980:
1977:
1972:
1968:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1935:
1931:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1872:
1866:
1845:shape operator
1832:regarded as a
1821:
1797:
1794:
1770:
1759:
1758:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1695:
1691:
1671:
1668:
1667:
1666:
1655:
1649:
1645:
1639:
1627:
1624:
1621:
1617:
1613:
1608:
1605:
1601:
1555:
1541:
1540:
1529:
1523:
1519:
1515:
1509:
1505:
1501:
1495:
1492:
1488:
1484:
1480:
1476:
1461:
1460:
1449:
1441:
1435:
1430:
1425:
1420:
1415:
1409:
1401:
1396:
1391:
1386:
1381:
1373:
1369:
1346:
1339:
1303:
1294:
1279:
1215:
1212:
1211:
1210:
1199:
1193:
1188:
1183:
1179:
1175:
1170:
1165:
1160:
1157:
1154:
1150:
1144:
1139:
1134:
1130:
1126:
1121:
1116:
1111:
1108:
1105:
1101:
1095:
1090:
1085:
1081:
1077:
1072:
1067:
1062:
1059:
1056:
1029:
1028:
1017:
1012:
1008:
1003:
1000:
995:
990:
987:
983:
978:
974:
969:
966:
961:
956:
953:
949:
944:
940:
935:
932:
927:
922:
919:
869:
868:
857:
851:
845:
842:
840:
837:
836:
833:
830:
828:
825:
824:
822:
801:
792:
784:
783:
772:
766:
762:
758:
754:
751:
748:
745:
741:
738:
734:
731:
728:
723:
719:
715:
711:
708:
704:
700:
685:
684:
673:
665:
659:
654:
649:
644:
639:
633:
625:
620:
615:
610:
605:
597:
593:
568:
559:
521:
510:
499:
488:
427:
424:
410:is tangent to
389:
388:
377:
371:
367:
363:
359:
356:
353:
350:
346:
343:
339:
336:
333:
328:
324:
320:
316:
303:quadratic form
287:
286:
275:
266:
261:
256:
252:
246:
243:
240:
237:
234:
231:
226:
221:
217:
211:
208:
205:
107:
104:
94:
91:
61:
57:
41:smooth surface
33:quadratic form
15:
13:
10:
9:
6:
4:
3:
2:
3102:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3067:
3065:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3031:
3029:
3027:
3023:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2993:
2991:
2989:
2985:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2958:Darboux frame
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2940:
2938:
2936:
2930:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2890:
2888:
2886:
2880:
2876:
2872:
2864:
2859:
2857:
2852:
2850:
2845:
2844:
2841:
2834:
2830:
2826:
2825:
2821:
2815:
2813:0-914098-72-1
2809:
2805:
2800:
2796:
2794:0-471-15732-5
2790:
2786:
2781:
2777:
2775:0-486-63433-7
2771:
2767:
2762:
2761:
2757:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2728:
2725:
2724:
2720:
2704:
2694:
2691:
2688:
2671:
2665:
2662:
2659:
2639:
2630:
2627:
2624:
2607:
2601:
2598:
2595:
2575:
2569:
2566:
2563:
2557:
2554:
2551:
2543:
2539:
2532:
2526:
2523:
2520:
2514:
2511:
2508:
2500:
2496:
2485:
2484:
2483:
2480:
2474:
2465:
2459:
2449:
2445:
2440:
2435:
2429:
2427:
2423:
2422:
2402:
2393:
2390:
2387:
2370:
2364:
2361:
2358:
2338:
2329:
2326:
2323:
2306:
2300:
2297:
2294:
2274:
2268:
2265:
2262:
2256:
2253:
2250:
2244:
2234:
2233:
2232:
2230:
2226:
2222:
2217:
2203:
2198:
2186:
2160:
2155:
2124:
2110:
2105:
2094:
2088:
2085:
2082:
2063:
2062:
2061:
2059:
2058:normal bundle
2055:
2047:
2045:
2043:
2038:
2032:
2030:
2026:
2021:
2016:
2011:
2007:
1985:
1981:
1975:
1970:
1962:
1959:
1953:
1950:
1944:
1941:
1938:
1933:
1922:
1919:
1916:
1910:
1907:
1901:
1895:
1889:
1883:
1880:
1877:
1856:
1855:
1854:
1851:
1846:
1841:
1839:
1838:metric tensor
1835:
1819:
1811:
1795:
1792:
1784:
1768:
1745:
1739:
1736:
1730:
1724:
1721:
1715:
1712:
1706:
1703:
1700:
1681:
1680:
1679:
1677:
1669:
1653:
1647:
1643:
1637:
1625:
1622:
1619:
1615:
1611:
1606:
1603:
1599:
1591:
1590:
1589:
1586:
1580:
1574:
1568:
1565:
1558:
1554:
1548:
1546:
1527:
1521:
1517:
1513:
1507:
1503:
1499:
1493:
1490:
1486:
1482:
1466:
1465:
1464:
1447:
1433:
1423:
1418:
1399:
1389:
1384:
1371:
1359:
1358:
1357:
1354:
1345:
1338:
1334:
1333:cross product
1329:
1323:
1316:
1312:
1302:
1293:
1282:
1278:
1272:
1266:
1261:
1256:
1250:
1243:
1239:
1235:
1231:
1225:
1222:
1197:
1191:
1181:
1173:
1168:
1158:
1155:
1152:
1148:
1142:
1132:
1124:
1119:
1109:
1106:
1103:
1099:
1093:
1083:
1075:
1070:
1060:
1057:
1054:
1047:
1046:
1045:
1042:
1038:
1034:
1015:
1006:
1001:
998:
988:
985:
981:
972:
967:
964:
954:
951:
947:
938:
933:
930:
920:
917:
910:
909:
908:
906:
901:
895:
889:
883:
879:
875:
855:
849:
843:
838:
831:
826:
820:
811:
810:
809:
804:
800:
795:
791:
770:
764:
760:
756:
752:
749:
746:
743:
739:
736:
732:
729:
726:
721:
717:
713:
709:
706:
690:
689:
688:
671:
657:
647:
642:
623:
613:
608:
595:
583:
582:
581:
578:
571:
567:
562:
558:
554:
553:cross product
549:
543:
536:
532:
524:
520:
513:
509:
502:
498:
491:
487:
481:
475:
469:
464:
459:
453:
446:
442:
438:
434:
425:
423:
420:
414:
407:
401:
395:
375:
369:
365:
361:
357:
354:
351:
348:
344:
341:
337:
334:
331:
326:
322:
318:
314:
307:
306:
305:
304:
298:
294:
273:
264:
259:
254:
250:
244:
241:
238:
235:
232:
229:
224:
219:
215:
209:
206:
203:
196:
195:
194:
192:
188:
183:
177:
172:
167:
162:
156:
149:
145:
141:
137:
132:
128:
123:
117:
113:
105:
99:
92:
90:
88:
84:
80:
76:
46:
42:
38:
37:tangent plane
34:
30:
26:
22:
2972:
2803:
2784:
2765:
2478:
2469:
2463:
2454:
2447:
2443:
2433:
2430:
2419:
2417:
2218:
2139:
2051:
2036:
2033:
2029:torsion-free
2019:
2013:denotes the
2009:
2005:
2000:
1849:
1847:(denoted by
1842:
1810:differential
1760:
1673:
1588:as follows:
1584:
1578:
1572:
1566:
1563:
1556:
1552:
1549:
1542:
1462:
1352:
1343:
1336:
1327:
1321:
1314:
1310:
1300:
1291:
1280:
1276:
1270:
1264:
1258:is a smooth
1254:
1248:
1241:
1237:
1233:
1229:
1226:
1220:
1217:
1040:
1030:
907:as follows:
899:
893:
887:
881:
877:
873:
870:
802:
798:
793:
789:
785:
686:
576:
569:
565:
560:
556:
547:
541:
534:
530:
522:
518:
511:
507:
500:
496:
489:
485:
479:
473:
467:
461:is a smooth
457:
451:
444:
440:
436:
432:
429:
418:
412:
405:
399:
393:
390:
296:
292:
288:
190:
181:
175:
165:
154:
147:
143:
139:
135:
121:
115:
109:
93:Surface in R
29:shape tensor
28:
24:
18:
3044:Cocurvature
2935:of surfaces
2873:defined in
2229:submanifold
2054:codimension
2042:orientation
905:dot product
83:submanifold
3064:Categories
2758:References
133:function,
106:Motivation
2893:Curvature
2885:of curves
2871:curvature
2768:. Dover.
2701:⟩
2643:⟨
2640:−
2637:⟩
2579:⟨
2573:⟩
2536:⟨
2530:⟩
2493:⟨
2400:⟩
2342:⟨
2339:−
2336:⟩
2278:⟨
2272:⟩
2242:⟨
2195:∇
2169:⊥
2152:∇
2119:⊥
2102:∇
1979:⟩
1967:∇
1957:⟨
1948:⟩
1930:∇
1926:⟨
1923:−
1914:⟩
1893:⟨
1820:ν
1796:ν
1783:Gauss map
1769:ν
1746:ν
1743:⟩
1725:ν
1719:⟨
1716:−
1648:γ
1638:γ
1626:β
1623:α
1607:β
1604:α
1522:β
1508:α
1494:β
1491:α
1424:×
1390:×
1182:⋅
1174:⋅
1159:−
1133:⋅
1125:⋅
1110:−
1084:⋅
1076:⋅
1061:−
1007:⋅
973:⋅
939:⋅
648:×
614:×
3049:Holonomy
2721:See also
1287:α = 1, 2
1252:, where
455:, where
169:and its
3090:Tensors
1781:is the
1037:Hessian
301:is the
161:tangent
35:on the
31:) is a
2810:
2791:
2772:
2223:, the
2140:where
2001:where
1785:, and
1761:where
1634:
1631:
1031:For a
23:, the
2831:from
2227:of a
127:Gauss
85:in a
39:of a
2808:ISBN
2789:ISBN
2770:ISBN
1808:the
1298:and
1227:Let
516:and
494:and
477:and
179:and
27:(or
2461:of
2219:In
2027:is
1812:of
1674:In
1274:by
1035:of
483:by
416:at
408:= 0
397:on
189:of
159:is
157:= 0
119:in
19:In
3066::
2482::
2428:.
1557:αÎČ
1547:.
1356::
1342:Ă
1285:,
1232:=
888:uv
880:,
876:,
797:,
580::
564:Ă
435:=
138:=
89:.
2862:e
2855:t
2848:v
2835:.
2816:.
2797:.
2778:.
2705:.
2698:)
2695:z
2692:,
2689:v
2686:(
2682:I
2676:I
2672:,
2669:)
2666:w
2663:,
2660:u
2657:(
2653:I
2647:I
2634:)
2631:w
2628:,
2625:v
2622:(
2618:I
2612:I
2608:,
2605:)
2602:z
2599:,
2596:u
2593:(
2589:I
2583:I
2576:+
2570:z
2567:,
2564:w
2561:)
2558:v
2555:,
2552:u
2549:(
2544:M
2540:R
2533:=
2527:z
2524:,
2521:w
2518:)
2515:v
2512:,
2509:u
2506:(
2501:N
2497:R
2479:M
2472:M
2470:R
2464:N
2457:N
2455:R
2450:)
2448:g
2446:,
2444:M
2442:(
2434:N
2403:.
2397:)
2394:z
2391:,
2388:v
2385:(
2381:I
2375:I
2371:,
2368:)
2365:w
2362:,
2359:u
2356:(
2352:I
2346:I
2333:)
2330:w
2327:,
2324:v
2321:(
2317:I
2311:I
2307:,
2304:)
2301:z
2298:,
2295:u
2292:(
2288:I
2282:I
2275:=
2269:z
2266:,
2263:w
2260:)
2257:v
2254:,
2251:u
2248:(
2245:R
2204:w
2199:v
2165:)
2161:w
2156:v
2148:(
2125:,
2115:)
2111:w
2106:v
2098:(
2095:=
2092:)
2089:w
2086:,
2083:v
2080:(
2076:I
2072:I
2037:n
2020:n
2010:w
2006:v
2003:â
1986:,
1982:n
1976:w
1971:v
1963:,
1960:n
1954:=
1951:n
1945:w
1942:,
1939:n
1934:v
1920:=
1917:n
1911:w
1908:,
1905:)
1902:v
1899:(
1896:S
1890:=
1887:)
1884:w
1881:,
1878:v
1875:(
1871:I
1865:I
1850:S
1793:d
1740:w
1737:,
1734:)
1731:v
1728:(
1722:d
1713:=
1710:)
1707:w
1704:,
1701:v
1698:(
1694:I
1690:I
1654:.
1644:n
1620:,
1616:r
1612:=
1600:b
1585:n
1579:S
1573:r
1567:u
1564:u
1553:b
1528:.
1518:u
1514:d
1504:u
1500:d
1487:b
1483:=
1479:I
1475:I
1448:.
1440:|
1434:2
1429:r
1419:1
1414:r
1408:|
1400:2
1395:r
1385:1
1380:r
1372:=
1368:n
1353:n
1347:2
1344:r
1340:1
1337:r
1328:S
1322:r
1317:)
1315:u
1313:,
1311:u
1309:(
1304:2
1301:r
1295:1
1292:r
1281:α
1277:r
1271:u
1265:r
1255:r
1249:R
1244:)
1242:u
1240:,
1238:u
1236:(
1234:r
1230:r
1221:S
1198:.
1192:v
1187:r
1178:H
1169:v
1164:r
1156:=
1153:N
1149:,
1143:v
1138:r
1129:H
1120:u
1115:r
1107:=
1104:M
1100:,
1094:u
1089:r
1080:H
1071:u
1066:r
1058:=
1055:L
1041:H
1016:.
1011:n
1002:v
999:v
994:r
989:=
986:N
982:,
977:n
968:v
965:u
960:r
955:=
952:M
948:,
943:n
934:u
931:u
926:r
921:=
918:L
900:S
894:r
882:N
878:M
874:L
856:.
850:]
844:N
839:M
832:M
827:L
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