1336:
1980:
1554:
535:
1887:
1044:
2075:
1828:
907:
2101:
689:
2010:
759:
1632:
645:
298:
205:
108:
1112:
1104:
998:
779:
616:
588:
428:
378:
338:
269:
229:
156:
136:
68:
1793:
1739:
1689:
1425:
1398:
975:
941:
875:
837:
475:
408:
48:
1499:
1451:
2030:
1759:
1709:
1655:
1603:
1583:
1471:
1371:
1084:
1064:
799:
713:
555:
448:
358:
318:
249:
176:
1892:
1714:
On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve:
1565:
The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold
1504:
2180:
2159:
480:
692:
2244:
1833:
75:
2201:
2039:
78:, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given
2196:
2134:
2191:
1003:
1804:
883:
2080:
2111:
654:
1985:
2104:
2033:
718:
1608:
1331:{\displaystyle k_{g}=\gamma ''(s)\cdot {\Big (}N(\gamma (s))\times \gamma '(s){\Big )}=\left\,,}
621:
274:
181:
84:
1799:
1453:, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius
2168:
1089:
983:
764:
601:
560:
413:
363:
323:
254:
214:
141:
121:
53:
590:, which explains why they appear to be curved in ambient space whenever the submanifold is.
1764:
1717:
1660:
1403:
1376:
946:
912:
846:
808:
453:
386:
26:
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is identically 1, independently of the direction considered. Great circles have curvature
8:
1476:
1430:
17:
208:
2249:
2220:
2015:
1744:
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1456:
1356:
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1049:
784:
698:
540:
433:
343:
303:
234:
161:
2217:
2176:
2155:
2147:
1975:{\displaystyle {\bar {\nabla }}_{T}T=\nabla _{T}T+({\bar {\nabla }}_{T}T)^{\perp }}
381:
1342:
2238:
2129:
410:), which depends only on the direction of the curve, and the curvature of
1889:. It splits into a tangent part and a normal part to the submanifold:
2225:
2124:
1798:
In general
Riemannian geometry, the derivative is computed using the
648:
477:), which is a second order quantity. The relation between these is
79:
71:
1657:
have zero geodesic curvature, which is equivalent to saying that
877:
on the normal bundle to the submanifold at the point considered.
1741:
only depends on the point on the submanifold and the direction
1400:
in three-dimensional
Euclidean space. The normal curvature of
1559:
557:
have zero geodesic curvature (they are "straight"), so that
2215:
805:
is the norm of the projection of the covariant derivative
340:
depends on two factors: the curvature of the submanifold
839:
on the tangent space to the submanifold. Conversely the
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2018:
1988:
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and it is different in general from the curvature of
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144:
124:
87:
56:
29:
2095:
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2032:(it is a particular case of Gauss equation in the
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2004:
1974:
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1822:
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1549:{\displaystyle k_{g}={\frac {\sqrt {1-r^{2}}}{r}}}
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1330:
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62:
42:
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1199:
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211:), geodesic curvature refers to the curvature of
2236:
1585:. It does not depend on the way the submanifold
880:If the ambient manifold is the euclidean space
530:{\displaystyle k={\sqrt {k_{g}^{2}+k_{n}^{2}}}}
76:1D curves on a 2D surface embedded in 3D space
2167:
2152:Differential Geometry of Curves and Surfaces
1341:where the square brackets denote the scalar
1027:
1007:
748:
728:
1982:. The tangent part is the usual derivative
1882:{\displaystyle DT/ds={\bar {\nabla }}_{T}T}
70:measures how far the curve is from being a
2189:
1560:Some results involving geodesic curvature
1324:
889:
2146:
2237:
1691:is orthogonal to the tangent space to
158:is restricted to lie on a submanifold
2216:
1106:, the geodesic curvature is given by
138:(see below). However, when the curve
2070:{\displaystyle \mathrm {I\!I} (T,T)}
1066:designates the unit normal field of
691:. Its curvature is the norm of the
13:
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2048:
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1990:
1944:
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1900:
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1811:
1309:
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14:
2261:
2209:
1039:{\displaystyle \|\gamma '(s)\|=1}
843:is the norm of the projection of
1823:{\displaystyle {\bar {\nabla }}}
909:, then the covariant derivative
902:{\displaystyle \mathbb {R} ^{n}}
2096:{\displaystyle \mathrm {I\!I} }
2064:
2052:
1963:
1947:
1937:
1903:
1864:
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1165:
1159:
1143:
1137:
1024:
1018:
631:
284:
191:
94:
1:
2140:
943:is just the usual derivative
684:{\displaystyle T=d\gamma /ds}
593:
537:. In particular geodesics on
2036:), while the normal part is
2005:{\displaystyle \nabla _{T}T}
7:
2197:Encyclopedia of Mathematics
2118:
754:{\displaystyle k=\|DT/ds\|}
651:, with unit tangent vector
10:
2266:
2190:Slobodyan, Yu.S. (2001) ,
1627:{\displaystyle {\bar {M}}}
1348:
640:{\displaystyle {\bar {M}}}
300:. The (ambient) curvature
293:{\displaystyle {\bar {M}}}
200:{\displaystyle {\bar {M}}}
103:{\displaystyle {\bar {M}}}
1830:of the ambient manifold:
271:in the ambient manifold
450:(the geodesic curvature
2135:Gauss–Codazzi equations
2105:second fundamental form
2034:Gauss-Codazzi equations
1501:and geodesic curvature
1099:{\displaystyle \gamma }
993:{\displaystyle \gamma }
774:{\displaystyle \gamma }
611:{\displaystyle \gamma }
583:{\displaystyle k=k_{n}}
423:{\displaystyle \gamma }
373:{\displaystyle \gamma }
333:{\displaystyle \gamma }
264:{\displaystyle \gamma }
224:{\displaystyle \gamma }
151:{\displaystyle \gamma }
131:{\displaystyle \gamma }
63:{\displaystyle \gamma }
2245:Geodesic (mathematics)
2169:Guggenheimer, Heinrich
2097:
2071:
2026:
2006:
1976:
1883:
1824:
1800:Levi-Civita connection
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2173:Differential Geometry
2148:do Carmo, Manfredo P.
2098:
2072:
2027:
2007:
1977:
1884:
1825:
1790:
1788:{\displaystyle DT/ds}
1756:
1736:
1734:{\displaystyle k_{n}}
1706:
1686:
1684:{\displaystyle DT/ds}
1652:
1629:
1600:
1580:
1551:
1496:
1468:
1448:
1422:
1420:{\displaystyle S^{2}}
1395:
1393:{\displaystyle S^{2}}
1368:
1333:
1101:
1081:
1061:
1041:
995:
972:
970:{\displaystyle dT/ds}
938:
936:{\displaystyle DT/ds}
904:
872:
870:{\displaystyle DT/ds}
834:
832:{\displaystyle DT/ds}
796:
776:
756:
710:
686:
642:
613:
585:
552:
532:
472:
470:{\displaystyle k_{g}}
445:
425:
405:
403:{\displaystyle k_{n}}
375:
355:
335:
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295:
266:
246:
226:
202:
173:
153:
133:
105:
65:
45:
43:{\displaystyle k_{g}}
2221:"Geodesic curvature"
2192:"Geodesic curvature"
2171:(1977), "Surfaces",
2112:Gauss–Bonnet theorem
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2040:
2016:
1986:
1893:
1834:
1805:
1765:
1745:
1718:
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1609:
1589:
1569:
1505:
1477:
1473:will have curvature
1457:
1431:
1404:
1377:
1357:
1113:
1090:
1070:
1050:
1004:
1000:is unit-speed, i.e.
984:
947:
913:
884:
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809:
785:
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719:
699:
693:covariant derivative
655:
622:
602:
561:
541:
481:
454:
434:
414:
387:
364:
360:in the direction of
344:
324:
304:
275:
255:
235:
215:
182:
162:
142:
122:
85:
54:
27:
1494:{\displaystyle 1/r}
1446:{\displaystyle k=1}
1373:be the unit sphere
524:
506:
74:. For example, for
18:Riemannian geometry
2218:Weisstein, Eric W.
2093:
2067:
2022:
2002:
1972:
1879:
1820:
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1751:
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1624:
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1417:
1390:
1363:
1328:
1096:
1076:
1056:
1036:
990:
967:
933:
899:
867:
829:
803:geodesic curvature
791:
771:
751:
705:
681:
647:, parametrized by
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209:curves on surfaces
197:
168:
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114:is just the usual
112:geodesic curvature
100:
60:
40:
22:geodesic curvature
2154:, Prentice-Hall,
2025:{\displaystyle M}
1950:
1906:
1867:
1817:
1754:{\displaystyle T}
1704:{\displaystyle M}
1650:{\displaystyle M}
1621:
1598:{\displaystyle M}
1578:{\displaystyle M}
1544:
1540:
1466:{\displaystyle r}
1366:{\displaystyle M}
1317:
1257:
1079:{\displaystyle M}
1059:{\displaystyle N}
794:{\displaystyle M}
708:{\displaystyle T}
634:
598:Consider a curve
550:{\displaystyle M}
525:
443:{\displaystyle M}
353:{\displaystyle M}
313:{\displaystyle k}
287:
244:{\displaystyle M}
194:
171:{\displaystyle M}
97:
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841:normal curvature
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382:normal curvature
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1279:
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1273:
1270:
1267:
1264:
1261:
1253:
1249:
1244:
1238:
1235:
1232:
1229:
1224:
1219:
1210:
1206:
1201:
1196:
1193:
1190:
1186:
1183:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1153:
1148:
1145:
1142:
1139:
1135:
1132:
1128:
1123:
1119:
1095:
1075:
1055:
1035:
1032:
1029:
1026:
1023:
1020:
1016:
1013:
1009:
989:
966:
963:
959:
955:
952:
932:
929:
925:
921:
918:
896:
891:
866:
863:
859:
855:
852:
828:
825:
821:
817:
814:
790:
770:
750:
747:
744:
740:
736:
733:
730:
727:
724:
704:
680:
677:
673:
669:
666:
663:
660:
633:
630:
618:in a manifold
607:
595:
592:
577:
573:
569:
566:
546:
522:
517:
513:
509:
504:
499:
495:
489:
486:
464:
460:
439:
419:
397:
393:
369:
349:
329:
309:
286:
283:
260:
240:
220:
193:
190:
167:
147:
127:
96:
93:
59:
37:
33:
9:
6:
4:
3:
2:
2262:
2251:
2248:
2246:
2243:
2242:
2240:
2228:
2227:
2222:
2219:
2214:
2213:
2203:
2199:
2198:
2193:
2188:
2184:
2182:0-486-63433-7
2178:
2174:
2170:
2166:
2163:
2161:0-13-212589-7
2157:
2153:
2149:
2145:
2144:
2136:
2133:
2131:
2130:Darboux frame
2128:
2126:
2123:
2122:
2113:
2109:
2106:
2061:
2058:
2055:
2035:
2019:
1999:
1994:
1967:
1959:
1954:
1934:
1931:
1926:
1918:
1915:
1910:
1876:
1871:
1854:
1851:
1848:
1844:
1840:
1837:
1801:
1797:
1782:
1779:
1775:
1771:
1768:
1761:, but not on
1748:
1726:
1722:
1713:
1698:
1678:
1675:
1671:
1667:
1664:
1644:
1637:Geodesics of
1636:
1615:
1592:
1572:
1564:
1563:
1557:
1541:
1535:
1531:
1527:
1524:
1518:
1513:
1509:
1488:
1484:
1480:
1460:
1440:
1437:
1434:
1412:
1408:
1385:
1381:
1360:
1346:
1344:
1325:
1320:
1313:
1300:
1294:
1283:
1274:
1268:
1262:
1259:
1251:
1247:
1233:
1227:
1222:
1208:
1204:
1191:
1184:
1181:
1177:
1168:
1162:
1156:
1146:
1140:
1133:
1130:
1126:
1121:
1117:
1109:
1108:
1107:
1093:
1073:
1053:
1033:
1030:
1021:
1014:
1011:
987:
978:
964:
961:
957:
953:
950:
930:
927:
923:
919:
916:
894:
878:
864:
861:
857:
853:
850:
842:
826:
823:
819:
815:
812:
804:
788:
768:
745:
742:
738:
734:
731:
725:
722:
702:
694:
678:
675:
671:
667:
664:
661:
658:
650:
628:
605:
591:
575:
571:
567:
564:
544:
520:
515:
511:
507:
502:
497:
493:
487:
484:
462:
458:
437:
417:
395:
391:
383:
367:
347:
327:
307:
281:
258:
238:
218:
210:
188:
165:
145:
125:
117:
113:
91:
81:
77:
73:
57:
35:
31:
23:
19:
2224:
2195:
2172:
2151:
2103:denotes the
1352:
1340:
979:
879:
840:
802:
597:
115:
111:
21:
15:
50:of a curve
2239:Categories
2141:References
594:Definition
207:(e.g. for
2250:Manifolds
2226:MathWorld
2202:EMS Press
2175:, Dover,
2125:Curvature
1991:∇
1968:⊥
1948:¯
1945:∇
1923:∇
1904:¯
1901:∇
1865:¯
1862:∇
1815:¯
1812:∇
1619:¯
1528:−
1295:γ
1269:γ
1228:γ
1182:γ
1178:×
1163:γ
1147:⋅
1131:γ
1094:γ
1028:‖
1012:γ
1008:‖
988:γ
769:γ
749:‖
729:‖
668:γ
649:arclength
632:¯
606:γ
418:γ
368:γ
328:γ
285:¯
259:γ
219:γ
192:¯
146:γ
126:γ
116:curvature
95:¯
58:γ
2150:(1976),
2119:See also
2077:, where
1605:sits in
1185:′
1134:″
1015:′
781:lies on
430:seen in
80:manifold
72:geodesic
1349:Example
2179:
2158:
1086:along
1046:, and
801:, the
110:, the
20:, the
761:. If
380:(the
2177:ISBN
2156:ISBN
2110:The
1353:Let
2012:in
980:If
695:of
320:of
231:in
178:of
118:of
16:In
2241::
2223:.
2200:,
2194:,
1556:.
1345:.
977:.
715::
2229:.
2205:.
2186:.
2114:.
2107:.
2090:I
2086:I
2065:)
2062:T
2059:,
2056:T
2053:(
2049:I
2045:I
2020:M
2000:T
1995:T
1964:)
1960:T
1955:T
1938:(
1935:+
1932:T
1927:T
1919:=
1916:T
1911:T
1877:T
1872:T
1855:=
1852:s
1849:d
1845:/
1841:T
1838:D
1795:.
1783:s
1780:d
1776:/
1772:T
1769:D
1749:T
1727:n
1723:k
1711:.
1699:M
1679:s
1676:d
1672:/
1668:T
1665:D
1645:M
1634:.
1616:M
1593:M
1573:M
1542:r
1536:2
1532:r
1525:1
1519:=
1514:g
1510:k
1489:r
1485:/
1481:1
1461:r
1441:1
1438:=
1435:k
1413:2
1409:S
1386:2
1382:S
1361:M
1326:,
1321:]
1314:s
1310:d
1304:)
1301:s
1298:(
1291:d
1284:,
1281:)
1278:)
1275:s
1272:(
1266:(
1263:N
1260:,
1252:2
1248:s
1243:d
1237:)
1234:s
1231:(
1223:2
1218:d
1209:[
1205:=
1200:)
1195:)
1192:s
1189:(
1175:)
1172:)
1169:s
1166:(
1160:(
1157:N
1152:(
1144:)
1141:s
1138:(
1127:=
1122:g
1118:k
1074:M
1054:N
1034:1
1031:=
1025:)
1022:s
1019:(
965:s
962:d
958:/
954:T
951:d
931:s
928:d
924:/
920:T
917:D
895:n
890:R
865:s
862:d
858:/
854:T
851:D
827:s
824:d
820:/
816:T
813:D
789:M
746:s
743:d
739:/
735:T
732:D
726:=
723:k
703:T
679:s
676:d
672:/
665:d
662:=
659:T
629:M
576:n
572:k
568:=
565:k
545:M
521:2
516:n
512:k
508:+
503:2
498:g
494:k
488:=
485:k
463:g
459:k
438:M
396:n
392:k
348:M
308:k
282:M
239:M
189:M
166:M
92:M
36:g
32:k
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