局部對(duì)稱(chēng)偽Riemann流形中的緊致極大類(lèi)時(shí)子流形

李 影,宋衛(wèi)東

(安徽師范大學(xué) 數(shù)學(xué)計(jì)算機(jī)科學(xué)學(xué)院,安徽 蕪湖 241003)

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研究簡(jiǎn)報(bào)

局部對(duì)稱(chēng)偽Riemann流形中的緊致極大類(lèi)時(shí)子流形

李 影,宋衛(wèi)東

(安徽師范大學(xué) 數(shù)學(xué)計(jì)算機(jī)科學(xué)學(xué)院,安徽 蕪湖 241003)

利用活動(dòng)標(biāo)架法,得到了局部對(duì)稱(chēng)偽Riemann流形中極大類(lèi)時(shí)子流形的一個(gè)Simons型積分不等式,以及該子流形成為全測(cè)地類(lèi)時(shí)子流形的關(guān)于其第二基本形式模長(zhǎng)平方的拼擠定理.

偽Riemann流形;局部對(duì)稱(chēng);極大類(lèi)時(shí)子流形;全測(cè)地類(lèi)時(shí)子流形

本文約定各類(lèi)指標(biāo)取值范圍如下:

1≤A,B,C,…≤n+p; 1≤i,j,k,…≤n;n+1≤α,β,γ,…≤n+p.

(1)

(2)

則有

(3)

(4)

類(lèi)似地,曲率張量場(chǎng)Kαijk的共變導(dǎo)數(shù)Kαijk,l定義為

限制到Mn上時(shí),有

(5)

(1-δ),A≠B;

(1-δ),A,B,C,D互不相同.

(6)

再由式(1),(2),(5),(6)得

(7)

(8)

(9)

(10)

由于(tr(HαHβ))p×p是實(shí)對(duì)稱(chēng)矩陣,因此選取法標(biāo)架場(chǎng){eα}可使之對(duì)角化,即

(11)

從而有

(12)

從而

(13)

由文獻(xiàn)[10]，顯然有

(14)

由式(7)～(14),有

(15)

由于Mn是緊致無(wú)邊的,根據(jù)Stocks定理,對(duì)式(15)兩邊積分得

證畢.

(16)

證明:由已知條件式(16)可知式(15)的右邊非負(fù),而Mn是緊致無(wú)邊的,由Hopf極大值原理可知S為常數(shù).從而式(15)左邊為零.因此,式(15)右邊也為零,即

結(jié)合已知條件可知S=0,故Mn是全測(cè)地類(lèi)時(shí)子流形.證畢.

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(責(zé)任編輯：趙立芹)

MaximumTimelikeSubmanifoldinaLocallySymmetricPseudo-RiemannianManifold

LI Ying,SONG Weidong

(CollegeofMathematicsandComputerScience,AnhuiNormalUniversity,Wuhu241003,AnhuiProvince,China)

Based on the moving frames,an integral inequality about maximal timelike submanifold was obtained in the locally symmetric pseudo-Riemannian manifold and a pinching theorem about the squared norm of the second fundamental form for the compact maximal timelike submanifold was gived in locally symmetric pseudo-Riemannian manifold.

pseudo-Riemannian manifold;locally symmetric;maximum timelike submanifold;totally geodesic timelike submanifold

10.13413/j.cnki.jdxblxb.2015.03.20

2014-09-15.

李 影(1991—),女,漢族,碩士研究生,從事微分幾何的研究,E-mail:909789714@qq.com.通信作者:宋衛(wèi)東(1958—),男,漢族,教授,從事微分幾何的研究,E-mail:swd56@sina.com.

國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào):11071005)和安徽省教育廳自然科學(xué)重點(diǎn)項(xiàng)目(批準(zhǔn)號(hào):KJ2010A125).

O186.12

：A

：1671-5489(2015)03-0457-04