Riemann-Finsler子流形幾何研究.pdf

1、飯里大學(xué)博士后工作報告Riemann—Finsler子流形幾何研究站點數(shù)學(xué)研究所專業(yè)基礎(chǔ)數(shù)掌作者吳炳燁日期2005年6月AbstractInthisreportweshallstudysubmanifoldsinfpseudo)RiemanniangeometryandFinslergeometryInChapter1weuseGaUSSmaptostudythetopologyvolumeanddiameterofsubmanifol

2、dsinasphereWjprovethatifthereexist1≥￡0andafixedpvectorasuchthattheGallSSmapgofanmdimensionalcompleteandconnectedsubmanifoldMiⅡS“psatisfies2E，thenMisdiffeomorphictoS“，andthevollimeanddiameterofMsatisfygnvol(S”)0andafixedu

3、nitsimple(n1)一vector口∈G暑】∞suchthattheGaussmapgofallndlmensiomdcompleteandconnectedspacehkesubmaififoldM“in島psatisfies《g，a)≤p，thenM“isdiffeomorphietoS“，anditsvolumesatisfiesvol(S”)／p_vol(M)≤礦vol(X)Wealsocharacterizethecas

4、ewheretheseinequalltiesbecomeequalitiesInChapter3weobtainalowerboundforthefirstDirichleteigenvalueofcompletespacelikehypersurfacesinLorentzianspaceintermsofmeancurvatureandthesquarelengthofthesecondfundamentalfc，rmThises

5、timateiSsharpfortotallyumbilicalhyperbohcspacesinLorentzianspaceWealsogetasufficientconditionforspacelikehypersurfacetohavezerofirsteigenvalueintermofGaussmapThenjnChapter4，wedealwiththeFinslergeometryofsubruanifoldswith

6、respecttogeneralFinslervolumeelement，ThekeyisthatShen’smethodstillworksindealinswithanyotherFinslervolumeelemeat，andweprovethatthereexistsnoclosedorientedminimalsubmanifoldinMinkowskispacewithre8pecttoanyFinslervolumeele

7、mentWealSOobtainanestimateofvolumegrowthforsubmanifoldsinspecialRandersspaceandthusprovidesanecessaryconditionforalhndersspacetobeminimallyimmersedintospecialRandersspaceFinallyinChapter5mobtainanextrinsicupperboundforth