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Berger's isoembolic inequality

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672: 244: 279:'s book "Manifolds all of whose geodesics are closed." At this stage, the isoembolic inequality appeared with a non-optimal constant. Sometimes Kazdan's inequality is called 713: 147: 737: 706: 564: 742: 47: 275:. The proof relies upon an analytic inequality proved by Kazdan. The original work of Berger and Kazdan appears in the appendices of 699: 611: 517: 470: 404: 732: 51: 595: 548: 547:. Cambridge Studies in Advanced Mathematics. Vol. 98 (Second edition of 1993 original ed.). Cambridge: 442: 648: 679: 438: 43: 39: 104: 17: 645: 607: 560: 513: 466: 400: 625: 599: 578: 552: 531: 505: 484: 458: 418: 392: 671: 621: 574: 527: 480: 414: 629: 617: 582: 570: 535: 523: 488: 476: 454: 422: 410: 388: 97: 683: 501: 250: 239:{\displaystyle \mathrm {vol} (M)\geq {\frac {c_{m}(\mathrm {inj} (M))^{m}}{\pi ^{m}}},} 509: 726: 446: 380: 69: 66: 450: 430: 276: 73: 437:. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 93. Appendices by 31: 462: 396: 556: 653: 59: 594:. Translations of Mathematical Monographs. Vol. 149. Providence, RI: 603: 265: 62: 271:
with its usual round metric. This result is known as Berger's
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with its usual "round" metric. The theorem is named after the
500:. Pure and Applied Mathematics. Vol. 115. Orlando, FL: 27:
Gives a lower bound on the volume of a Riemannian manifold
687: 150: 238: 72:, who derived it from an inequality proved by 707: 8: 42:that gives a lower bound on the volume of a 435:Manifolds all of whose geodesics are closed 714: 700: 545:Riemannian geometry. A modern introduction 225: 214: 190: 181: 174: 151: 149: 141:-dimensional sphere of radius one. Then 385:A panoramic view of Riemannian geometry 292: 363: 335: 331: 323: 307: 303: 299: 103:-dimensional Riemannian manifold with 351: 339: 327: 311: 7: 668: 666: 649:"Berger-Kazdan comparison theorem" 498:Eigenvalues in Riemannian geometry 197: 194: 191: 158: 155: 152: 137:denote the volume of the standard 48:necessary and sufficient condition 25: 670: 122:denote the Riemannian volume of 18:Berger–Kazdan comparison theorem 738:Theorems in Riemannian geometry 211: 207: 201: 187: 168: 162: 36:Berger's isoembolic inequality 1: 596:American Mathematical Society 510:10.1016/s0079-8169(08)x6051-9 686:. You can help Knowledge by 759: 665: 549:Cambridge University Press 743:Riemannian geometry stubs 463:10.1007/978-3-642-61876-5 397:10.1007/978-3-642-18245-7 557:10.1017/CBO9780511616822 281:Berger–Kazdan inequality 80:Statement of the theorem 590:Sakai, Takashi (1996). 50:for the manifold to be 733:Geometric inequalities 682:-related article is a 543:Chavel, Isaac (2006). 496:Chavel, Isaac (1984). 240: 445:, L. BĂ©rard-Bergery, 342:, Proposition VI.2.2. 273:isoembolic inequality 241: 264:is isometric to the 148: 680:Riemannian geometry 592:Riemannian geometry 453:. Berlin–New York: 338:, Theorem VII.2.1; 310:, Theorem VII.2.2; 44:Riemannian manifold 40:Riemannian geometry 646:Weisstein, Eric W. 236: 105:injectivity radius 695: 694: 604:10.1090/mmono/149 566:978-0-521-61954-7 443:J.-P. Bourguignon 314:, Theorem VI.2.1. 231: 46:and also gives a 16:(Redirected from 750: 716: 709: 702: 674: 667: 659: 658: 633: 586: 539: 492: 439:D. B. A. Epstein 431:Besse, Arthur L. 426: 367: 361: 355: 349: 343: 321: 315: 306:, Theorem V.22; 297: 268: 263: 245: 243: 242: 237: 232: 230: 229: 220: 219: 218: 200: 186: 185: 175: 161: 140: 136: 125: 121: 113: 102: 95: 57: 21: 758: 757: 753: 752: 751: 749: 748: 747: 723: 722: 721: 720: 663: 644: 643: 640: 614: 589: 567: 542: 520: 495: 473: 455:Springer-Verlag 429: 407: 389:Springer-Verlag 379: 371: 370: 362: 358: 350: 346: 334:, Theorem V.1; 322: 318: 302:, Theorem 148; 298: 294: 289: 266: 253: 221: 210: 177: 176: 146: 145: 138: 135: 127: 123: 115: 107: 100: 85: 82: 55: 38:is a result in 28: 23: 22: 15: 12: 11: 5: 756: 754: 746: 745: 740: 735: 725: 724: 719: 718: 711: 704: 696: 693: 692: 675: 661: 660: 639: 638:External links 636: 635: 634: 612: 587: 565: 540: 518: 502:Academic Press 493: 471: 427: 405: 381:Berger, Marcel 369: 368: 366:, Theorem V.1. 356: 344: 330:, Appendix E; 316: 291: 290: 288: 285: 251:if and only if 249:with equality 247: 246: 235: 228: 224: 217: 213: 209: 206: 203: 199: 196: 193: 189: 184: 180: 173: 170: 167: 164: 160: 157: 154: 131: 81: 78: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 755: 744: 741: 739: 736: 734: 731: 730: 728: 717: 712: 710: 705: 703: 698: 697: 691: 689: 685: 681: 676: 673: 669: 664: 656: 655: 650: 647: 642: 641: 637: 631: 627: 623: 619: 615: 613:0-8218-0284-4 609: 605: 601: 597: 593: 588: 584: 580: 576: 572: 568: 562: 558: 554: 550: 546: 541: 537: 533: 529: 525: 521: 519:0-12-170640-0 515: 511: 507: 503: 499: 494: 490: 486: 482: 478: 474: 472:3-540-08158-5 468: 464: 460: 456: 452: 448: 444: 440: 436: 432: 428: 424: 420: 416: 412: 408: 406:3-540-65317-1 402: 398: 394: 390: 386: 382: 378: 377: 376: 375: 365: 360: 357: 354:, Appendix D. 353: 348: 345: 341: 337: 333: 329: 326:, Lemma 158; 325: 320: 317: 313: 309: 305: 301: 296: 293: 286: 284: 282: 278: 274: 270: 261: 257: 252: 233: 226: 222: 215: 204: 182: 178: 171: 165: 144: 143: 142: 134: 130: 119: 111: 106: 99: 93: 89: 79: 77: 75: 71: 70:Marcel Berger 68: 67:mathematician 64: 61: 53: 49: 45: 41: 37: 33: 19: 688:expanding it 677: 662: 652: 591: 544: 497: 451:J. L. Kazdan 434: 384: 373: 372: 359: 347: 319: 295: 280: 277:Arthur Besse 272: 259: 255: 248: 132: 128: 117: 109: 91: 87: 83: 74:Jerry Kazdan 35: 29: 364:Chavel 1984 336:Chavel 2006 332:Chavel 1984 324:Berger 2003 308:Chavel 2006 304:Chavel 1984 300:Berger 2003 60:dimensional 32:mathematics 727:Categories 630:0886.53002 583:1099.53001 536:0551.53001 489:0387.53010 423:1038.53002 387:. Berlin: 352:Besse 1978 340:Sakai 1996 328:Besse 1978 312:Sakai 1996 287:References 654:MathWorld 447:M. Berger 223:π 172:≥ 52:isometric 433:(1978). 383:(2003). 126:and let 622:1390760 575:2229062 528:0768584 481:0496885 415:2002701 269:-sphere 258:,  90:,  54:to the 628:  620:  610:  581:  573:  563:  534:  526:  516:  487:  479:  469:  421:  413:  403:  374:Books. 114:. 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Index

Berger–Kazdan comparison theorem
mathematics
Riemannian geometry
Riemannian manifold
necessary and sufficient condition
isometric
dimensional
sphere
mathematician
Marcel Berger
Jerry Kazdan
closed
injectivity radius
if and only if
m-sphere
Arthur Besse
Berger 2003
Chavel 1984
Chavel 2006
Sakai 1996
Berger 2003
Besse 1978
Chavel 1984
Chavel 2006
Sakai 1996
Besse 1978
Chavel 1984
Berger, Marcel
Springer-Verlag
doi

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