關(guān)于半歐氏空間之間調(diào)和同態(tài)的一些結(jié)論.pdf

1、摘要本文研究半歐氏空間RmRnT9之間的調(diào)和態(tài)射。通過推廣[12和【14」中構(gòu)造歐氏空間之間調(diào)和態(tài)射的方法，氏空間之間兩個(gè)相應(yīng)的定理(如下)，并給出證明我們得出半歐。同時(shí)，我們也給出半歐氏空間之間二次調(diào)和態(tài)射的一些有趣例子。定理2.3設(shè)0:提升(1):RxRI么中是調(diào)和態(tài)射RmaRnrs是齊二次多項(xiàng)式調(diào)和態(tài)射，笠閨鱺￡k=1Rn定義為，(xy)一(、雪器y..ax.。這里RmxRTn具有乘積度量形式0的完全器。i)0.Y，那skdxk.

2、dxk.定理2.4設(shè)0i釁、珊和42釁毋是兩個(gè)調(diào)和態(tài)射，它們的直和O1.0:定義為(0i(D02)(xy)=01(二)02(y)，乘積流形AlrxNqp具有積度量，那么01.02是調(diào)和態(tài)射。關(guān)鍵詞:半黎曼流形，調(diào)和映射，調(diào)和態(tài)射。IntroductionThenotionsofharmonicmorphismsandhorizontallyweaklyconformalmapsbetweenRiemannianmanifoldswere

3、studiedinthecontextofdiferentialgeometryinthelate70softhelastcentury.Aharmonicmorphismischaracterizedasahorizontallyweaklyconformalandharmonicmapindependentlyin[8]and[10].Foradetailedaccountonharmonicmapsandharmonicmorph

4、ismsbetweenRiemannianmanifoldswereferto[2][4][5][6]andthereferencestherein.Recently，thecorrespondingnotionsforsemiRiemannianmanifoldshavebeenstudiedbyParmar[15].WerefertoONeill[11]concerningsemiRiemannianmanifolds(wheret

5、hemetrictensormaybeindefiniteandhencetheLaplaceBetramioperatormaynotbeelliptic).Thisthesisisdividedintotwosections.InSection1weintroduceandrecallsomefundamentalconceptsandfactsconcerningsemiRiemannianmanifoldsandharmonic

6、mapsharmonicmorphismsandhorizontalweaklyconformalmapsbetweenthesespaces.InSection2weproveourmaintheoremsandgiveaexampleofnontrivialharmonicmorphismQ:R璧R要andshowthatitscompleteliftisagainaquadraticharmonicmorphisn蠶:R2x形理T