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The assumption on "nonempty" has meaning: a fixed point theorem often states the set of fixed point is an
Hadamard space. The main content of such an assertion is that the set is nonempty.
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658:. The analog holds for a Hadamard space: a complete, connected metric space which is locally isometric to a Hadamard space has an Hadamard space as its
1819:
2014:
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828:: in a gas of hard balls, is there a uniform bound on the number of collisions? The solution begins by constructing a configuration space for the
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Every bounded subset of a
Hadamard space is contained in the smallest closed ball (which is the same as the closure of its convex hull). If
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if the above inequality is equality. A flat
Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a
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Burago D., Ferleger S. Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. of Math. 147 (1998), 695-708
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24:; that is, the middle one in the picture. In fact, any complete metric space where a triangle is hyperbolic is an Hadamard space.
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is a bounded subset of a metric space, then the center of the closed ball of the minimum radius containing it is called the
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832:, obtained by joining together copies of corresponding billiard table, which turns out to be a Hadamard space.
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The geometry of
Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of
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850: â complete, simply-connected Riemannian manifold with nonpositive sectional curvature everywhere
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910:, IRMA Lectures in Mathematics and Theoretical Physics, vol. 6 (Second ed.),
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805:. Important examples of Hadamard manifolds are simply connected nonpositively curved
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280:{\displaystyle d(z,m)^{2}+{d(x,y)^{2} \over 4}\leq {d(z,x)^{2}+d(z,y)^{2} \over 2}.}
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Applications of
Hadamard spaces are not restricted to geometry. In 1998,
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44:. In the literature they are also equivalently defined as complete
15:
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499:. In a Hadamard space, any two points can be joined by a unique
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953:
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The basic result for a non-positively curved manifold is the
492:
is an
Hadamard space if and only if it is a Hilbert space.
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In a
Hilbert space, the above inequality is equality (with
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A Hadamard space is defined to be a nonempty complete
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Pages displaying wikidata descriptions as a fallback
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Metric spaces, convexity and non-positive curvature
1309:Spectral theory of ordinary differential equations
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484:), and in general an Hadamard space is said to be
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927:Burago, Dmitri; Yuri Burago, and Sergei Ivanov.
662:. Its variant applies for non-positively curved
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897:Bridson, Martin R.; Haefliger, André (1999),
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844: â Type of metric space in mathematics
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879:A Course in Metric Geometry, p. 334.
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1262:Group algebra of a locally compact group
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931:. American Mathematical Society. (1984)
899:Metric spaces of non-positive curvature
863:
429:{\displaystyle d(x,m)=d(y,m)=d(x,y)/2.}
939:Notes on the Theory of Hadamard Spaces
584:of a Hadamard space leaving invariant
40:, is a non-linear generalization of a
797:, that is, complete simply-connected
7:
20:In an Hadamard space, a triangle is
503:between them; in particular, it is
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563:
14:
1580:Compact operator on Hilbert space
1418:
1417:
1344:Topological quantum field theory
669:Examples of Hadamard spaces are
906:Papadopoulos, Athanase (2014),
649:BruhatâTits fixed point theorem
1732:Differentiable/Smooth manifold
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1140:Uniform boundedness principle
912:European Mathematical Society
945:Notes on Alexandrov Geometry
786:{\displaystyle 2pq\geq p+q,}
55:such that, given any points
2438:Classification of manifolds
929:A Course in Metric Geometry
2588:
1549:Hilbert projection theorem
1283:Invariant subspace problem
627:fixes the circumcenter of
118:such that for every point
2514:over commutative algebras
1528:CauchyâSchwarz inequality
1413:
1003:
745:{\displaystyle p,q\geq 3}
477:{\displaystyle m=(x+y)/2}
2230:Riemann curvature tensor
1252:Spectrum of a C*-algebra
309:is then the midpoint of
1349:Noncommutative geometry
681:(for example, complete
656:CartanâHadamard theorem
620:{\displaystyle \Gamma }
569:{\displaystyle \Gamma }
2022:Manifold with boundary
1737:Differential structure
1405:TomitaâTakesaki theory
1380:Approximation property
1324:Calculus of variations
824:to solve a problem in
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507:. Quite generally, if
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1559:Polarization identity
1502:Orthogonal complement
1400:BanachâMazur distance
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711:{\displaystyle (p,q)}
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98:there exists a point
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2169:Covariant derivative
1720:Topological manifold
1533:Riesz representation
1488:L-semi-inner product
1145:Kakutani fixed-point
1130:Riesz representation
799:Riemannian manifolds
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2557:Functional analysis
2203:Exterior derivative
1805:AtiyahâSinger index
1754:Riemannian manifold
1554:Parseval's identity
1523:Bessel's inequality
1329:Functional calculus
1288:Mahler's conjecture
1267:Von Neumann algebra
981:Functional analysis
826:dynamical billiards
803:sectional curvature
2562:Geometric topology
2509:Secondary calculus
2463:Singularity theory
2418:Parallel transport
2186:De Rham cohomology
1825:Generalized Stokes
1354:Riemann hypothesis
1053:Topological vector
795:Hadamard manifolds
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600:{\displaystyle B,}
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549:{\displaystyle B.}
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345:{\displaystyle y:}
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134:{\displaystyle z,}
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91:{\displaystyle y,}
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2191:Differential form
1845:Whitney embedding
1779:Differential form
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1610:Sesquilinear form
1563:Parallelogram law
1507:Orthonormal basis
1431:
1430:
1334:Integral operator
1111:
1110:
921:978-3-03719-132-3
848:Hadamard manifold
640:{\displaystyle B}
520:{\displaystyle B}
497:rigidity theorems
322:{\displaystyle x}
302:{\displaystyle m}
272:
207:
111:{\displaystyle m}
68:{\displaystyle x}
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2536:Stratified space
2494:Fréchet manifold
2208:Interior product
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1497:Prehilbert space
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1257:Operator algebra
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830:dynamical system
807:symmetric spaces
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38:Jacques Hadamard
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2479:Banach manifold
2472:Generalizations
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2174:Cotangent space
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1595:HilbertâSchmidt
1585:Densely defined
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1373:Advanced topics
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1160:GelfandâNaimark
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822:CAT(0) geometry
801:of nonpositive
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666:. (cf. Lurie.)
660:universal cover
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2567:Hilbert spaces
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2032:Parallelizable
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1912:Lie derivative
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1907:Integral curve
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1868:Diffeomorphism
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1713:Basic concepts
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1590:Hermitian form
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1472:Basic concepts
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1465:Hilbert spaces
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1390:Choquet theory
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1244:
1242:Banach algebra
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1228:
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1203:
1198:
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1177:
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1168:
1167:
1165:BanachâAlaoglu
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1132:
1127:
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1119:
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1095:
1093:Locally convex
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854:
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837:
834:
818:Serge Ferleger
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764:
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707:
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671:Hilbert spaces
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107:
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84:
64:
36:, named after
34:Hadamard space
13:
10:
9:
6:
4:
3:
2:
2584:
2573:
2572:Metric spaces
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2554:
2552:
2537:
2534:
2532:
2531:Supermanifold
2529:
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2524:
2522:
2519:
2515:
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2510:
2507:
2505:
2502:
2500:
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2495:
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2490:
2487:
2485:
2482:
2480:
2477:
2476:
2474:
2470:
2464:
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
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2436:
2435:
2433:
2429:
2419:
2416:
2414:
2411:
2409:
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2404:
2401:
2399:
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2379:
2376:
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2356:
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2351:
2348:
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2336:
2333:
2331:
2328:
2326:
2322:
2318:
2316:
2313:
2311:
2308:
2306:
2302:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2276:
2273:
2271:
2268:
2267:
2265:
2263:
2259:
2253:
2252:Wedge product
2250:
2248:
2245:
2241:
2238:
2237:
2236:
2233:
2231:
2228:
2224:
2221:
2220:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2197:
2196:Vector-valued
2194:
2193:
2192:
2189:
2187:
2184:
2180:
2177:
2176:
2175:
2172:
2170:
2167:
2165:
2162:
2161:
2159:
2155:
2149:
2146:
2144:
2141:
2139:
2136:
2132:
2129:
2128:
2127:
2126:Tangent space
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2108:
2106:
2102:
2099:
2097:
2093:
2087:
2084:
2082:
2078:
2074:
2072:
2069:
2067:
2063:
2059:
2055:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2016:
2013:
2012:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1957:
1953:
1951:
1947:
1943:
1941:
1938:
1937:
1935:
1929:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1900:
1898:
1895:
1891:
1890:in Lie theory
1888:
1887:
1886:
1883:
1881:
1878:
1874:
1871:
1870:
1869:
1866:
1864:
1861:
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1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1802:
1800:
1797:
1793:Main results
1791:
1785:
1782:
1780:
1777:
1775:
1774:Tangent space
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1752:
1750:
1747:
1743:
1740:
1738:
1735:
1734:
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1730:
1726:
1723:
1722:
1721:
1718:
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1711:
1706:
1702:
1695:
1690:
1688:
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1676:
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1660:
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1627:
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1608:
1606:
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1598:
1596:
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1577:
1575:
1571:
1564:
1560:
1557:
1555:
1552:
1550:
1547:
1546:
1544:
1542:Other results
1540:
1534:
1531:
1529:
1526:
1524:
1521:
1520:
1518:
1514:
1508:
1505:
1503:
1500:
1498:
1494:
1493:Hilbert space
1491:
1489:
1485:
1484:Inner product
1482:
1480:
1477:
1476:
1474:
1470:
1466:
1459:
1454:
1452:
1447:
1445:
1440:
1439:
1436:
1424:
1416:
1415:
1412:
1406:
1403:
1401:
1398:
1396:
1395:Weak topology
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1377:
1375:
1371:
1364:
1360:
1357:
1355:
1352:
1350:
1347:
1345:
1342:
1340:
1337:
1335:
1332:
1330:
1327:
1325:
1322:
1320:
1319:Index theorem
1317:
1315:
1312:
1310:
1307:
1305:
1302:
1301:
1299:
1295:
1289:
1286:
1284:
1281:
1280:
1278:
1276:Open problems
1274:
1268:
1265:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1239:
1237:
1233:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1199:
1197:
1194:
1192:
1189:
1187:
1184:
1182:
1179:
1178:
1176:
1172:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1122:
1120:
1118:
1114:
1104:
1101:
1099:
1096:
1094:
1091:
1088:
1084:
1080:
1077:
1075:
1072:
1070:
1067:
1066:
1064:
1060:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1005:
1002:
999:
995:
990:
986:
982:
975:
970:
968:
963:
961:
956:
955:
952:
946:
942:
940:
936:
933:
930:
926:
923:
917:
913:
909:
904:
900:
895:
894:
885:
882:
876:
873:
867:
864:
857:
849:
846:
843:
840:
839:
835:
833:
831:
827:
823:
819:
815:
814:Dmitri Burago
810:
808:
804:
800:
796:
780:
777:
774:
771:
768:
765:
762:
759:
739:
736:
733:
730:
727:
719:
702:
699:
696:
684:
680:
676:
675:Poincaré disc
672:
667:
665:
661:
657:
652:
650:
634:
594:
591:
583:
579:
543:
540:
532:
531:
514:
506:
502:
498:
493:
491:
485:
471:
467:
460:
457:
454:
448:
445:
436:
423:
419:
412:
409:
406:
400:
397:
391:
388:
385:
379:
376:
370:
367:
364:
358:
339:
336:
316:
296:
287:
274:
269:
263:
255:
252:
249:
243:
240:
235:
227:
224:
221:
215:
209:
204:
198:
190:
187:
184:
178:
172:
167:
159:
156:
153:
147:
128:
125:
105:
85:
82:
62:
54:
49:
47:
46:CAT(0) spaces
43:
42:Hilbert space
39:
35:
31:
23:
18:
2458:Moving frame
2453:Morse theory
2443:Gauge theory
2235:Tensor field
2164:Closed/Exact
2143:Vector field
2111:Distribution
2052:Hypercomplex
2047:Quaternionic
1784:Vector field
1742:Smooth atlas
1657:
1648:
1644:
1640:
1636:
1605:Self-adjoint
1516:Main results
1385:Balanced set
1359:Distribution
1297:Applications
1150:KreinâMilman
1135:Closed graph
928:
907:
898:
884:
875:
866:
842:CAT(k) space
811:
668:
653:
648:
530:circumcenter
528:
505:contractible
494:
490:normed space
437:
288:
53:metric space
50:
33:
27:
2403:Levi-Civita
2393:Generalized
2365:Connections
2315:Lie algebra
2247:Volume form
2148:Vector flow
2121:Pushforward
2116:Lie bracket
2015:Lie algebra
1980:G-structure
1769:Pushforward
1749:Submanifold
1615:Trace class
1314:Heat kernel
1304:Hardy space
1211:Trace class
1125:HahnâBanach
1087:Topological
935:Jacob Lurie
677:, complete
2551:Categories
2526:Stratifold
2484:Diffeology
2280:Associated
2081:Symplectic
2066:Riemannian
1995:Hyperbolic
1922:Submersion
1830:HopfâRinow
1764:Submersion
1759:Smooth map
1247:C*-algebra
1062:Properties
901:, Springer
858:References
679:real trees
582:isometries
289:The point
22:hyperbolic
2408:Principal
2383:Ehresmann
2340:Subbundle
2330:Principal
2305:Fibration
2285:Cotangent
2157:Covectors
2010:Lie group
1990:Hermitian
1933:manifolds
1902:Immersion
1897:Foliation
1835:Noether's
1820:Frobenius
1815:De Rham's
1810:Darboux's
1701:Manifolds
1221:Unbounded
1216:Transpose
1174:Operators
1103:Separable
1098:Reflexive
1083:Algebraic
1069:Barrelled
769:≥
737:≥
664:orbifolds
615:Γ
564:Γ
210:≤
2504:Orbifold
2499:K-theory
2489:Diffiety
2213:Pullback
2027:Oriented
2005:Kenmotsu
1985:Hadamard
1931:Types of
1880:Geodesic
1705:Glossary
1629:Examples
1423:Category
1235:Algebras
1117:Theorems
1074:Complete
1043:Schwartz
989:glossary
836:See also
501:geodesic
30:geometry
2448:History
2431:Related
2345:Tangent
2323:)
2303:)
2270:Adjoint
2262:Bundles
2240:density
2138:Torsion
2104:Vectors
2096:Tensors
2079:)
2064:)
2060:,
2058:Pseudoâ
2037:Poisson
1970:Finsler
1965:Fibered
1960:Contact
1958:)
1950:Complex
1948:)
1917:Section
1643:) with
1620:Unitary
1479:Adjoint
1226:Unitary
1206:Nuclear
1191:Compact
1186:Bounded
1181:Adjoint
1155:Minâmax
1048:Sobolev
1033:Nuclear
1023:Hilbert
1018:Fréchet
983: (
576:is the
2413:Vector
2398:Koszul
2378:Cartan
2373:Affine
2355:Vector
2350:Tensor
2335:Spinor
2325:Normal
2321:Stable
2275:Affine
2179:bundle
2131:bundle
2077:Almost
2000:KĂ€hler
1956:Almost
1946:Almost
1940:Closed
1840:Sard's
1796:(list)
1600:Normal
1201:Normal
1038:Orlicz
1028:Hölder
1008:Banach
997:Spaces
985:topics
918:
718:-space
673:, the
2521:Sheaf
2295:Fiber
2071:Rizza
2042:Prime
1873:Local
1863:Curve
1725:Atlas
1651:<â
1013:Besov
820:used
720:with
607:then
578:group
32:, an
2388:Form
2290:Dual
2223:flow
2086:Tame
2062:Subâ
1975:Flat
1855:Maps
1573:Maps
1495:and
1486:and
1361:(or
1079:Dual
916:ISBN
816:and
793:and
752:and
651:).
486:flat
329:and
75:and
2310:Jet
685:),
580:of
533:of
28:In
2553::
2301:Co
987:â
937::
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809:.
424:2.
48:.
2319:(
2299:(
2075:(
2056:(
1954:(
1944:(
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1561:(
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647:(
635:B
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592:B
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541:B
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455:x
452:(
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420:/
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410:,
407:x
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392:m
389:,
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380:d
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371:m
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270:2
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