1016:, then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature is indeed 0. It is also possible to construct a torus by identifying opposite sides of a square, in which case the Riemannian metric on the torus is flat and has constant curvature 0, again resulting in total curvature 0. It is not possible to specify a Riemannian metric on the torus with everywhere positive or everywhere negative Gaussian curvature.
129:
1772:
32:
979:, its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will. The theorem states, somewhat surprisingly, that the total integral of all curvatures will remain the same, no matter how the deforming is done. So for instance if you have a sphere with a "dent", then its
1216:
In the case of the plane (where the
Gaussian curvature is 0 and geodesics are straight lines), we recover the familiar formula for the sum of angles in an ordinary triangle. On the standard sphere, where the curvature is everywhere 1, we see that the angle sum of geodesic triangles is always bigger
1520:
998:
without boundary, with curvature 0 and with Euler characteristic 1: the Gauss–Bonnet formula does not work. It holds true however for the compact closed unit disc, which also has Euler characteristic 1, because of the added boundary integral with value
785:
1372:
1102:
339:
567:
On the other hand, suppose we flatten the hemisphere to make it into a disk. This transformation is a homeomorphism, so the Euler characteristic is still 1. However, on the left hand side of the theorem we now have
882:
1245:, the area of a triangle is proportional to the amount by which its interior angles fail to add up to 180°, or equivalently by the (inverse) amount by which its exterior angles fail to add up to 360°.
1691:
1211:
688:
940:
562:
633:
484:
818:
517:
1229:
A number of earlier results in spherical geometry and hyperbolic geometry, discovered over the preceding centuries, were subsumed as special cases of Gauss–Bonnet.
592:
696:
1256:– the amount by which its interior angles add up to more than 180°, which is equal to the amount by which its exterior angles add up to less than 360°.
1515:{\displaystyle \sum _{v\,\in \,\operatorname {int} M}{\bigl (}6-\chi (v){\bigr )}+\sum _{v\,\in \,\partial M}{\bigl (}3-\chi (v){\bigr )}=6\chi (M),\ }
1030:
171:
is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries.
3225:
249:
1010:
has Euler characteristic 0, so its total curvature must also be zero. If the torus carries the ordinary
Riemannian metric from its embedding in
2416:
2120:
1285:
1280:
3220:
96:
68:
968:
of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and
3268:
2507:
2283:
2133:
2531:
49:
75:
2726:
1547:
Similar formulas can be obtained for 2-dimensional pseudo-manifold when we replace triangles with higher polygons. For polygons of
823:
2313:
2288:
82:
2596:
2175:
2058:
1797:
2822:
1720:
1616:
1156:
2875:
2403:
1711:
adjacent points on the surface. This is the simplest formula of Gauss–Bonnet theorem in three-dimensional digital space.
64:
3159:
1809:
2148:
2113:
1959:
638:
115:
2924:
1324:. This is the special case of Gauss–Bonnet, where the curvature is concentrated at discrete points (the vertices).
2907:
2516:
1814:
1753:
898:
1913:
1877:
53:
3119:
2526:
2078:
1937:
1292:
is the polyhedral analog: it states that the sum of the defect at all the vertices of a polyhedron which is
820:
almost everywhere along the border, which is a geodesic triangle. But we have three right-angle corners, so
3104:
2827:
2601:
2210:
2106:
1335:, and Gauss–Bonnet for measures generalizes both Gauss–Bonnet for smooth manifolds and Descartes' theorem.
2068:
132:
An example of a complex region where Gauss–Bonnet theorem can apply. Shows the sign of geodesic curvature.
89:
3149:
2073:
1760:
1735:
414:
as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the
168:
522:
3154:
3124:
2832:
2788:
2769:
2536:
2480:
2231:
3273:
2691:
2556:
2360:
1343:
There are several combinatorial analogs of the Gauss–Bonnet theorem. We state the following one. Let
892:
The theorem applies in particular to compact surfaces without boundary, in which case the integral
3076:
2941:
2633:
2475:
2153:
1742:
20:
945:
can be omitted. It states that the total
Gaussian curvature of such a closed surface is equal to 2
2773:
2743:
2667:
2657:
2613:
2443:
2396:
2339:
1268:
42:
3114:
2733:
2628:
2541:
2448:
2370:
2303:
2298:
2236:
2195:
1602:
we must replace 3 and 6 in the formula above with 2 and 4, respectively. More specifically, if
1238:
1141:
minus the total curvature within the triangle. Since the turning angle at a corner is equal to
597:
448:
2365:
2013:
2763:
2758:
1328:
790:
489:
137:
1792:
The theorem can be used directly as a system to control sculpture - for example, in work by
3094:
3032:
2880:
2584:
2574:
2546:
2521:
2431:
2200:
1301:
426:
372:
179:
175:
8:
3232:
2914:
2792:
2777:
2706:
2465:
2257:
2226:
2214:
2185:
2168:
2129:
1965:
1331:, rather than as a function, Descartes' theorem is Gauss–Bonnet where the curvature is a
1260:
1253:
1242:
965:
950:
571:
199:
3205:
2092:
780:{\displaystyle \int _{M}KdA={\frac {1}{R^{2}}}{\frac {4\pi R^{2}}{8}}={\frac {\pi }{2}}}
519:, because the boundary is the equator and the equator is a geodesic of the sphere. Then
421:
Many standard proofs use the theorem of turning tangents, which states roughly that the
3174:
3129:
3026:
2897:
2701:
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2252:
2221:
1249:
233:
214:
164:
153:
2711:
2262:
2018:
Proceedings of
Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture
3109:
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2716:
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2054:
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1919:
1909:
1901:
1883:
1873:
1865:
1834:
1724:
1130:
The geodesic curvature the bordering geodesics is 0, and the Euler characteristic of
1112:
1941:
987:(the Euler characteristic of a sphere being 2), no matter how big or deep the dent.
3247:
3041:
2996:
2919:
2890:
2748:
2681:
2676:
2671:
2661:
2453:
2436:
2334:
2329:
2205:
2158:
1992:
1947:
1785:
1752:
A far-reaching generalization that includes all the abovementioned theorems is the
1746:
1607:
1332:
390:
1127:
formed by the inside of that triangle and the piecewise boundary of the triangle.
3190:
3099:
2929:
2885:
2651:
2163:
2098:
1348:
995:
980:
1997:
1980:
3056:
2981:
2951:
2849:
2842:
2782:
2753:
2623:
2618:
2579:
2046:
1838:
1793:
990:
Compactness of the surface is of crucial importance. Consider for instance the
445:. Its Euler characteristic is 1. On the left hand side of the theorem, we have
422:
349:
3262:
3242:
3066:
3061:
3046:
3036:
2986:
2963:
2837:
2797:
2738:
2686:
2485:
2190:
1599:
693:
Finally, take a sphere octant, also homeomorphic to the previous cases. Then
195:
1887:
1097:{\displaystyle \int _{T}K=2\pi -\sum \alpha -\int _{\partial T}\kappa _{g},}
3169:
3164:
3006:
2973:
2946:
2854:
2495:
1923:
1293:
334:{\displaystyle \int _{M}K\,dA+\int _{\partial M}k_{g}\,ds=2\pi \chi (M),\,}
1951:
1137:
Hence the sum of the turning angles of the geodesic triangle is equal to 2
3012:
3001:
2958:
2859:
2460:
1317:
is the genus, meaning "number of holes"), then the sum of the defect is
3237:
3195:
3021:
2934:
2566:
2470:
2381:
2308:
1289:
128:
1981:"Digital topological method for computing genus and the Betti numbers"
1771:
3051:
3016:
2721:
2608:
1780:
991:
149:
31:
3215:
3210:
3200:
2591:
2412:
2180:
2051:
Introduction to Tensor
Analysis and the Calculus of Moving Surfaces
1120:
953:
compact surfaces without boundary, the Euler characteristic equals
157:
418:
by which the smooth portions turn at the corners of the boundary.
1775:
Sculpture made from flat materials using the Gauss–Bonnet
Theorem
1119:
to be a simply connected region whose boundary consists of three
1525:
where the first sum ranges over the vertices in the interior of
2807:
635:, because a circumference is not a geodesic of the plane. Then
1149:
The sum of interior angles of a geodesic triangle is equal to
1759:
A generalization to 2-manifolds that need not be compact is
1007:
949:
times the Euler characteristic of the surface. Note that for
415:
1551:
vertices, we must replace 3 and 6 in the formula above with
877:{\displaystyle \int _{\partial M}k_{g}ds={\frac {3\pi }{2}}}
1145:
minus the interior angle, we can rephrase this as follows:
441:
is the northern hemisphere cut out from a sphere of radius
1789:, two characters discuss the derivation of this theorem.
1707:
indicates the number of surface-points each of which has
19:"Gauss–Bonnet" redirects here. Not to be confused with
178:, who developed a version but never published it, and
1619:
1375:
1362:
denote the number of triangles containing the vertex
1159:
1033:
901:
826:
793:
699:
641:
600:
574:
525:
492:
451:
252:
1686:{\displaystyle g=1+{\frac {M_{5}+2M_{6}-M_{3}}{8}},}
1529:, the second sum is over the boundary vertices, and
1206:{\displaystyle \sum (\pi -\alpha )=\pi +\int _{T}K.}
163:
In the simplest application, the case of a triangle
1153:plus the total curvature enclosed by the triangle:
56:. Unsourced material may be challenged and removed.
2128:
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3260:
1024:Sometimes the Gauss–Bonnet formula is stated as
683:{\displaystyle \int _{\partial M}k_{g}ds=2\pi }
1745:can also be seen as a generalization of GB to
2397:
2114:
1860:
1858:
1483:
1458:
1427:
1402:
1906:Differential geometry of curves and surfaces
1734:-dimensional generalization of GB (also see
1908:. Upper Saddle River, N.J.: Prentice-Hall.
935:{\displaystyle \int _{\partial M}k_{g}\,ds}
148:) is a fundamental formula which links the
2404:
2390:
2284:Fundamental theorem of Riemannian geometry
2121:
2107:
1894:
1855:
1286:Descartes' theorem on total angular defect
1281:Descartes' theorem on total angular defect
356:is the line element along the boundary of
1996:
1844:(Interview). Interviewed by Allyn Jackson
1447:
1443:
1388:
1384:
925:
330:
299:
266:
116:Learn how and when to remove this message
2411:
2045:
1978:
1900:
1864:
1770:
1300:. More generally, if the polyhedron has
182:, who published a special case in 1848.
127:
2011:
1338:
3261:
1979:Chen, Li; Rong, Yongwu (August 2010).
1123:. We can then apply GB to the surface
2385:
2102:
1936:
1833:
1798:University of Arkansas Honors College
1766:
975:If one bends and deforms the surface
1263:, conversely is proportional to its
54:adding citations to reliable sources
25:
432:
13:
2039:
1839:"Interview with Shiing-Shen Chern"
1714:
1448:
1252:is proportional to its excess, by
1073:
907:
832:
647:
557:{\displaystyle \int _{M}KdA=2\pi }
281:
14:
3285:
3269:Theorems in differential geometry
2086:
1598:, respectively. For example, for
1115:. Here we define a "triangle" on
393:, then we interpret the integral
1224:
1019:
30:
16:Theorem in differential geometry
1540:is the Euler characteristic of
888:Interpretation and significance
41:needs additional citations for
2444:Differentiable/Smooth manifold
2005:
1972:
1930:
1827:
1503:
1497:
1478:
1472:
1422:
1416:
1175:
1163:
972:counts the number of handles.
324:
318:
1:
1985:Topology and Its Applications
1820:
136:In the mathematical field of
2211:Raising and lowering indices
1274:
1232:
185:
7:
3150:Classification of manifolds
2074:Encyclopedia of Mathematics
1998:10.1016/j.topol.2010.04.006
1902:do Carmo, Manfredo Perdigão
1866:do Carmo, Manfredo Perdigão
1815:Atiyah–Singer index theorem
1803:
1754:Atiyah–Singer index theorem
1327:Thinking of curvature as a
174:The theorem is named after
10:
3290:
2232:Pseudo-Riemannian manifold
1810:Chern–Gauss–Bonnet theorem
1606:is a closed 2-dimensional
1347:be a finite 2-dimensional
1278:
18:
3226:over commutative algebras
3183:
3142:
3075:
2972:
2868:
2815:
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2642:
2565:
2504:
2424:
2361:Geometrization conjecture
2348:
2322:
2276:
2245:
2141:
1796:in the collection of the
628:{\displaystyle k_{g}=1/R}
479:{\displaystyle K=1/R^{2}}
2942:Riemann curvature tensor
2014:"Gauss-Bonnet Sculpting"
2012:Harriss, Edmund (2020).
1761:Cohn-Vossen's inequality
1736:Chern–Weil homomorphism
1610:, the genus turns out
1269:Johann Heinrich Lambert
1243:hyperbolic trigonometry
813:{\displaystyle k_{g}=0}
512:{\displaystyle k_{g}=0}
2734:Manifold with boundary
2449:Differential structure
2371:Uniformization theorem
2304:Nash embedding theorem
2237:Riemannian volume form
2196:Levi-Civita connection
2069:"Gauss–Bonnet theorem"
1872:. Boston: Birkhäuser.
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1239:spherical trigonometry
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65:"Gauss–Bonnet theorem"
1952:10.1201/9780203912669
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1006:As an application, a
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138:differential geometry
131:
2881:Covariant derivative
2432:Topological manifold
2294:Gauss–Bonnet theorem
2201:Covariant derivative
2095:at Wolfram Mathworld
2093:Gauss–Bonnet Theorem
1966:Taylor & Francis
1743:Riemann–Roch theorem
1617:
1373:
1339:Combinatorial analog
1302:Euler characteristic
1267:, as established by
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1031:
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791:
697:
639:
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572:
523:
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373:Euler characteristic
352:of the surface, and
250:
180:Pierre Ossian Bonnet
176:Carl Friedrich Gauss
146:Gauss–Bonnet formula
142:Gauss–Bonnet theorem
50:improve this article
21:Gauss–Bonnet gravity
2915:Exterior derivative
2517:Atiyah–Singer index
2466:Riemannian manifold
2366:Poincaré conjecture
2227:Riemannian manifold
2215:Musical isomorphism
2130:Riemannian geometry
1870:Riemannian geometry
1261:hyperbolic triangle
587:{\displaystyle K=0}
200:Riemannian manifold
3221:Secondary calculus
3175:Singularity theory
3130:Parallel transport
2898:De Rham cohomology
2537:Generalized Stokes
2356:General relativity
2299:Hopf–Rinow theorem
2246:Types of manifolds
2222:Parallel transport
1943:The Shape of Space
1835:Chern, Shiing-Shen
1777:
1767:In popular culture
1683:
1512:
1455:
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1296:to the sphere is 4
1250:spherical triangle
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234:geodesic curvature
215:Gaussian curvature
156:to its underlying
134:
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2903:Differential form
2557:Whitney embedding
2491:Differential form
2379:
2378:
2060:978-1-4614-7866-9
1991:(12): 1931–1936.
1938:Weeks, Jeffrey R.
1837:(March 4, 1998).
1747:complex manifolds
1725:Shiing-Shen Chern
1678:
1511:
1435:
1376:
1113:geodesic triangle
872:
775:
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169:sum of its angles
126:
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100:
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3274:Riemann surfaces
3248:Stratified space
3206:Fréchet manifold
2920:Interior product
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391:piecewise smooth
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382:If the boundary
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3191:Banach manifold
3184:Generalizations
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2930:Ricci curvature
2886:Cotangent space
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2597:Exponential map
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2323:Generalizations
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2176:Exponential map
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2047:Grinfeld, Pavel
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447:
446:
442:
438:
435:
429:is exactly ±1.
409:
401:
394:
383:
376:
361:
357:
353:
350:element of area
345:
293:
289:
280:
276:
257:
253:
251:
248:
247:
237:
230:
222:
218:
210:
203:
191:
188:
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
12:
11:
5:
3287:
3277:
3276:
3271:
3254:
3253:
3251:
3250:
3245:
3240:
3235:
3230:
3229:
3228:
3218:
3213:
3208:
3203:
3198:
3193:
3187:
3185:
3181:
3180:
3178:
3177:
3172:
3167:
3162:
3157:
3152:
3146:
3144:
3140:
3139:
3136:
3135:
3133:
3132:
3127:
3122:
3117:
3112:
3107:
3102:
3097:
3092:
3087:
3081:
3079:
3073:
3072:
3070:
3069:
3064:
3059:
3054:
3049:
3044:
3039:
3029:
3024:
3019:
3009:
3004:
2999:
2994:
2989:
2984:
2978:
2976:
2970:
2969:
2967:
2966:
2961:
2956:
2955:
2954:
2944:
2939:
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2937:
2927:
2922:
2917:
2912:
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2900:
2895:
2894:
2893:
2883:
2878:
2872:
2870:
2866:
2865:
2863:
2862:
2857:
2852:
2847:
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2845:
2835:
2830:
2825:
2819:
2817:
2810:
2804:
2803:
2801:
2800:
2795:
2785:
2780:
2766:
2761:
2756:
2751:
2746:
2744:Parallelizable
2741:
2736:
2731:
2730:
2729:
2719:
2714:
2709:
2704:
2699:
2694:
2689:
2684:
2679:
2674:
2664:
2654:
2648:
2646:
2640:
2639:
2637:
2636:
2631:
2626:
2624:Lie derivative
2621:
2619:Integral curve
2616:
2611:
2606:
2605:
2604:
2594:
2589:
2588:
2587:
2580:Diffeomorphism
2577:
2571:
2569:
2563:
2562:
2560:
2559:
2554:
2549:
2544:
2539:
2534:
2529:
2524:
2519:
2513:
2511:
2502:
2501:
2499:
2498:
2493:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2457:
2456:
2451:
2441:
2440:
2439:
2428:
2426:
2425:Basic concepts
2422:
2421:
2409:
2408:
2401:
2394:
2386:
2377:
2376:
2374:
2373:
2368:
2363:
2358:
2352:
2350:
2346:
2345:
2343:
2342:
2340:Sub-Riemannian
2337:
2332:
2326:
2324:
2320:
2319:
2317:
2316:
2311:
2306:
2301:
2296:
2291:
2286:
2280:
2278:
2274:
2273:
2271:
2270:
2265:
2260:
2255:
2249:
2247:
2243:
2242:
2240:
2239:
2234:
2229:
2224:
2219:
2218:
2217:
2208:
2203:
2198:
2188:
2183:
2178:
2173:
2172:
2171:
2166:
2161:
2156:
2145:
2143:
2142:Basic concepts
2139:
2138:
2126:
2125:
2118:
2111:
2103:
2097:
2096:
2088:
2087:External links
2085:
2084:
2083:
2065:
2059:
2041:
2038:
2035:
2034:
2004:
1971:
1960:
1940:(2001-12-12).
1929:
1914:
1893:
1878:
1854:
1825:
1824:
1822:
1819:
1818:
1817:
1812:
1805:
1802:
1794:Edmund Harriss
1768:
1765:
1716:
1713:
1701:
1694:
1693:
1682:
1677:
1671:
1667:
1663:
1658:
1654:
1650:
1647:
1642:
1638:
1631:
1628:
1625:
1622:
1600:quadrilaterals
1523:
1522:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1485:
1480:
1477:
1474:
1471:
1468:
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1460:
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1450:
1446:
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1438:
1434:
1429:
1424:
1421:
1418:
1415:
1412:
1409:
1404:
1397:
1394:
1391:
1387:
1383:
1379:
1340:
1337:
1279:Main article:
1276:
1273:
1259:The area of a
1248:The area of a
1234:
1231:
1226:
1223:
1214:
1213:
1202:
1199:
1194:
1190:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1105:
1104:
1093:
1088:
1084:
1078:
1075:
1071:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1041:
1037:
1021:
1018:
992:open unit disc
943:
942:
931:
928:
922:
918:
912:
909:
905:
889:
886:
871:
867:
864:
858:
855:
852:
847:
843:
837:
834:
830:
809:
806:
801:
797:
774:
771:
766:
761:
755:
751:
747:
744:
734:
730:
726:
721:
718:
715:
712:
707:
703:
679:
676:
673:
670:
667:
662:
658:
652:
649:
645:
624:
620:
616:
613:
608:
604:
583:
580:
577:
553:
550:
547:
544:
541:
538:
533:
529:
508:
505:
500:
496:
473:
469:
464:
460:
457:
454:
434:
431:
423:winding number
405:
396:
342:
341:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
296:
292:
286:
283:
279:
275:
272:
269:
265:
260:
256:
226:
202:with boundary
187:
184:
124:
123:
38:
36:
29:
15:
9:
6:
4:
3:
2:
3286:
3275:
3272:
3270:
3267:
3266:
3264:
3249:
3246:
3244:
3243:Supermanifold
3241:
3239:
3236:
3234:
3231:
3227:
3224:
3223:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3188:
3186:
3182:
3176:
3173:
3171:
3168:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3147:
3145:
3141:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3082:
3080:
3078:
3074:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3034:
3030:
3028:
3025:
3023:
3020:
3018:
3014:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2979:
2977:
2975:
2971:
2965:
2964:Wedge product
2962:
2960:
2957:
2953:
2950:
2949:
2948:
2945:
2943:
2940:
2936:
2933:
2932:
2931:
2928:
2926:
2923:
2921:
2918:
2916:
2913:
2909:
2908:Vector-valued
2906:
2905:
2904:
2901:
2899:
2896:
2892:
2889:
2888:
2887:
2884:
2882:
2879:
2877:
2874:
2873:
2871:
2867:
2861:
2858:
2856:
2853:
2851:
2848:
2844:
2841:
2840:
2839:
2838:Tangent space
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2820:
2818:
2814:
2811:
2809:
2805:
2799:
2796:
2794:
2790:
2786:
2784:
2781:
2779:
2775:
2771:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2750:
2747:
2745:
2742:
2740:
2737:
2735:
2732:
2728:
2725:
2724:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2673:
2669:
2665:
2663:
2659:
2655:
2653:
2650:
2649:
2647:
2641:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2610:
2607:
2603:
2602:in Lie theory
2600:
2599:
2598:
2595:
2593:
2590:
2586:
2583:
2582:
2581:
2578:
2576:
2573:
2572:
2570:
2568:
2564:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2538:
2535:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2514:
2512:
2509:
2505:Main results
2503:
2497:
2494:
2492:
2489:
2487:
2486:Tangent space
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2455:
2452:
2450:
2447:
2446:
2445:
2442:
2438:
2435:
2434:
2433:
2430:
2429:
2427:
2423:
2418:
2414:
2407:
2402:
2400:
2395:
2393:
2388:
2387:
2384:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2353:
2351:
2347:
2341:
2338:
2336:
2333:
2331:
2328:
2327:
2325:
2321:
2315:
2314:Schur's lemma
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2290:
2289:Gauss's lemma
2287:
2285:
2282:
2281:
2279:
2275:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2250:
2248:
2244:
2238:
2235:
2233:
2230:
2228:
2225:
2223:
2220:
2216:
2212:
2209:
2207:
2204:
2202:
2199:
2197:
2194:
2193:
2192:
2191:Metric tensor
2189:
2187:
2186:Inner product
2184:
2182:
2179:
2177:
2174:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2151:
2150:
2147:
2146:
2144:
2140:
2135:
2131:
2124:
2119:
2117:
2112:
2110:
2105:
2104:
2101:
2094:
2091:
2090:
2080:
2076:
2075:
2070:
2066:
2062:
2056:
2052:
2048:
2044:
2043:
2023:
2019:
2015:
2008:
1999:
1994:
1990:
1986:
1982:
1975:
1967:
1963:
1961:9780203912669
1957:
1953:
1949:
1946:. CRC Press.
1945:
1944:
1939:
1933:
1925:
1921:
1917:
1911:
1907:
1903:
1897:
1889:
1885:
1881:
1875:
1871:
1867:
1861:
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1816:
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1807:
1801:
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1795:
1790:
1788:
1787:
1782:
1773:
1764:
1762:
1757:
1755:
1750:
1748:
1744:
1739:
1737:
1732:
1727:1945) is the
1726:
1722:
1721:Chern theorem
1712:
1704:
1700:
1680:
1675:
1669:
1665:
1661:
1656:
1652:
1648:
1645:
1640:
1636:
1629:
1626:
1623:
1620:
1613:
1612:
1611:
1609:
1601:
1591:
1583:
1567:
1559:
1545:
1537:
1533:
1506:
1500:
1494:
1491:
1488:
1475:
1469:
1466:
1463:
1451:
1444:
1440:
1436:
1432:
1419:
1413:
1410:
1407:
1395:
1392:
1389:
1385:
1381:
1377:
1369:
1368:
1367:
1359:
1355:
1350:
1336:
1334:
1330:
1325:
1322:
1311:
1307:
1303:
1295:
1291:
1287:
1282:
1272:
1270:
1266:
1262:
1257:
1255:
1251:
1246:
1244:
1240:
1230:
1225:Special cases
1222:
1200:
1197:
1192:
1188:
1184:
1181:
1178:
1172:
1169:
1166:
1160:
1148:
1147:
1146:
1135:
1128:
1122:
1114:
1091:
1086:
1082:
1076:
1069:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1039:
1035:
1027:
1026:
1025:
1020:For triangles
1017:
1014:
1009:
1004:
997:
993:
988:
982:
973:
967:
958:
952:
929:
926:
920:
916:
910:
903:
895:
894:
893:
885:
869:
865:
862:
856:
853:
850:
845:
841:
835:
828:
807:
804:
799:
795:
772:
769:
764:
759:
753:
749:
745:
742:
732:
728:
724:
719:
716:
713:
710:
705:
701:
691:
677:
674:
671:
668:
665:
660:
656:
650:
643:
622:
618:
614:
611:
606:
602:
581:
578:
575:
565:
551:
548:
545:
542:
539:
536:
531:
527:
506:
503:
498:
494:
471:
467:
462:
458:
455:
452:
430:
428:
424:
419:
417:
412:
408:
404:
400:
392:
387:
380:
374:
368:
364:
351:
327:
321:
315:
312:
309:
306:
303:
300:
294:
290:
284:
277:
273:
270:
267:
263:
258:
254:
246:
245:
244:
241:
235:
229:
225:
216:
207:
201:
197:
183:
181:
177:
172:
170:
166:
161:
159:
155:
151:
147:
143:
139:
130:
120:
117:
109:
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: –
66:
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
3170:Moving frame
3165:Morse theory
3155:Gauge theory
2947:Tensor field
2876:Closed/Exact
2855:Vector field
2823:Distribution
2764:Hypercomplex
2759:Quaternionic
2496:Vector field
2454:Smooth atlas
2349:Applications
2293:
2277:Main results
2072:
2053:. Springer.
2050:
2026:. Retrieved
2021:
2017:
2007:
1988:
1984:
1974:
1964:– via
1942:
1932:
1905:
1896:
1869:
1846:. Retrieved
1829:
1791:
1784:
1778:
1758:
1751:
1740:
1730:
1718:
1702:
1698:
1695:
1589:
1581:
1565:
1557:
1546:
1535:
1531:
1524:
1357:
1353:
1342:
1326:
1320:
1309:
1305:
1294:homeomorphic
1284:
1264:
1258:
1247:
1236:
1228:
1215:
1136:
1129:
1106:
1023:
1012:
1005:
989:
974:
956:
944:
891:
692:
566:
436:
427:Jordan curve
420:
410:
406:
402:
398:
385:
381:
366:
362:
343:
239:
227:
223:
205:
189:
173:
162:
145:
141:
135:
112:
106:October 2020
103:
93:
86:
79:
72:
60:
48:Please help
43:verification
40:
3115:Levi-Civita
3105:Generalized
3077:Connections
3027:Lie algebra
2959:Volume form
2860:Vector flow
2833:Pushforward
2828:Lie bracket
2727:Lie algebra
2692:G-structure
2481:Pushforward
2461:Submanifold
3263:Categories
3238:Stratifold
3196:Diffeology
2992:Associated
2793:Symplectic
2778:Riemannian
2707:Hyperbolic
2634:Submersion
2542:Hopf–Rinow
2476:Submersion
2471:Smooth map
2309:Ricci flow
2258:Hyperbolic
2028:2020-11-17
1915:0132125897
1879:0817634908
1848:2019-07-22
1821:References
1290:polyhedron
951:orientable
221:, and let
165:on a plane
76:newspapers
3120:Principal
3095:Ehresmann
3052:Subbundle
3042:Principal
3017:Fibration
2997:Cotangent
2869:Covectors
2722:Lie group
2702:Hermitian
2645:manifolds
2614:Immersion
2609:Foliation
2547:Noether's
2532:Frobenius
2527:De Rham's
2522:Darboux's
2413:Manifolds
2253:Hermitian
2206:Signature
2169:Sectional
2149:Curvature
2079:EMS Press
2024:: 137–144
1783:'s novel
1781:Greg Egan
1662:−
1495:χ
1470:χ
1467:−
1449:∂
1445:∈
1437:∑
1414:χ
1411:−
1393:
1386:∈
1378:∑
1275:Polyhedra
1233:Triangles
1189:∫
1182:π
1173:α
1170:−
1167:π
1161:∑
1134:being 1.
1121:geodesics
1083:κ
1074:∂
1070:∫
1066:−
1063:α
1060:∑
1057:−
1054:π
1036:∫
908:∂
904:∫
866:π
833:∂
829:∫
770:π
746:π
702:∫
678:π
648:∂
644:∫
552:π
528:∫
316:χ
313:π
282:∂
278:∫
255:∫
186:Statement
150:curvature
3216:Orbifold
3211:K-theory
3201:Diffiety
2925:Pullback
2739:Oriented
2717:Kenmotsu
2697:Hadamard
2643:Types of
2592:Geodesic
2417:Glossary
2268:Kenmotsu
2181:Geodesic
2134:Glossary
2049:(2014).
1904:(1976).
1888:24667701
1868:(1992).
1804:See also
1786:Diaspora
960:, where
437:Suppose
360:. Here,
190:Suppose
158:topology
3160:History
3143:Related
3057:Tangent
3035:)
3015:)
2982:Adjoint
2974:Bundles
2952:density
2850:Torsion
2816:Vectors
2808:Tensors
2791:)
2776:)
2772:,
2770:Pseudo−
2749:Poisson
2682:Finsler
2677:Fibered
2672:Contact
2670:)
2662:Complex
2660:)
2629:Section
2335:Hilbert
2330:Finsler
2081:, 2001
1924:1529515
1723:(after
1595:
1577:
1571:
1554:
1366:. Then
1329:measure
1313:(where
1308:= 2 − 2
964:is the
371:is the
348:is the
243:. Then
232:be the
213:be the
196:compact
154:surface
90:scholar
3125:Vector
3110:Koszul
3090:Cartan
3085:Affine
3067:Vector
3062:Tensor
3047:Spinor
3037:Normal
3033:Stable
2987:Affine
2891:bundle
2843:bundle
2789:Almost
2712:Kähler
2668:Almost
2658:Almost
2652:Closed
2552:Sard's
2508:(list)
2263:Kähler
2159:Scalar
2154:tensor
2057:
1958:
1922:
1912:
1886:
1876:
1696:where
1510:
1351:. Let
1265:defect
1107:where
787:. Now
416:angles
344:where
209:. Let
167:, the
140:, the
92:
85:
78:
71:
63:
3233:Sheaf
3007:Fiber
2783:Rizza
2754:Prime
2585:Local
2575:Curve
2437:Atlas
2164:Ricci
1842:(PDF)
1288:of a
1217:than
1111:is a
1008:torus
966:genus
955:2 − 2
425:of a
194:is a
152:of a
97:JSTOR
83:books
3100:Form
3002:Dual
2935:flow
2798:Tame
2774:Sub−
2687:Flat
2567:Maps
2055:ISBN
2022:2020
1956:ISBN
1920:OCLC
1910:ISBN
1884:OCLC
1874:ISBN
1741:The
1719:The
1574:and
1241:and
983:is 4
594:and
486:and
144:(or
69:news
3022:Jet
1993:doi
1989:157
1948:doi
1779:In
1738:).
1592:− 2
1568:− 2
1390:int
1237:In
389:is
375:of
236:of
217:of
52:by
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2077:,
2071:,
2020:.
2016:.
1987:.
1983:.
1954:.
1918:.
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1857:^
1800:.
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884:.
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411:ds
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354:ds
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2063:.
2031:.
2001:.
1995::
1968:.
1950::
1926:.
1890:.
1851:.
1731:n
1729:2
1709:i
1703:i
1699:M
1681:,
1676:8
1670:3
1666:M
1657:6
1653:M
1649:2
1646:+
1641:5
1637:M
1630:+
1627:1
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1621:g
1604:M
1590:n
1586:/
1582:n
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1566:n
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1498:(
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1476:v
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1420:v
1417:(
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1319:2
1315:g
1310:g
1306:χ
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1198:K
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1164:(
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1143:π
1139:π
1132:T
1125:T
1117:M
1109:T
1092:,
1087:g
1077:T
1051:2
1048:=
1045:K
1040:T
1013:R
1001:π
999:2
985:π
977:M
970:g
962:g
957:g
947:π
930:s
927:d
921:g
917:k
911:M
870:2
863:3
857:=
854:s
851:d
846:g
842:k
836:M
808:0
805:=
800:g
796:k
773:2
765:=
760:8
754:2
750:R
743:4
733:2
729:R
725:1
720:=
717:A
714:d
711:K
706:M
675:2
672:=
669:s
666:d
661:g
657:k
651:M
623:R
619:/
615:1
612:=
607:g
603:k
582:0
579:=
576:K
549:2
546:=
543:A
540:d
537:K
532:M
507:0
504:=
499:g
495:k
472:2
468:R
463:/
459:1
456:=
453:K
443:R
439:M
407:g
403:k
399:M
397:∂
395:∫
386:M
384:∂
377:M
369:)
367:M
365:(
363:χ
358:M
328:,
325:)
322:M
319:(
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304:s
301:d
295:g
291:k
285:M
274:+
271:A
268:d
264:K
259:M
240:M
238:∂
228:g
224:k
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211:K
206:M
204:∂
192:M
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113:(
108:)
104:(
94:·
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46:.
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