復(fù)射影空間中具有常數(shù)量曲率的全實(shí)子流形

劉敏,宋衛(wèi)東

(安徽師范大學(xué)數(shù)計(jì)學(xué)院,安徽 蕪湖 241000)

復(fù)射影空間中具有常數(shù)量曲率的全實(shí)子流形

劉敏,宋衛(wèi)東

(安徽師范大學(xué)數(shù)計(jì)學(xué)院,安徽 蕪湖 241000)

通過活動(dòng)標(biāo)架法,研究了復(fù)射影空間中具有常數(shù)量曲率的全實(shí)子流形,得到其成為全臍子流形的剛性定理,并推廣了相關(guān)結(jié)果.

復(fù)射影空間;全實(shí)子流形;數(shù)量曲率;全臍

1 引言

設(shè)CPn+p是具有Fubini-Study度量的復(fù)n+p維復(fù)射影空間,全純截面曲率為常數(shù)4.設(shè)J為CPn+p的復(fù)結(jié)構(gòu),Mn為CPn+p的實(shí)n維子流形.如果Mn上每點(diǎn)切空間被J變換到自身,則稱Mn是CPn+p的全純子流形.與此相反,若Mn上每點(diǎn)的切空間被J變換到該點(diǎn)法空間,則稱Mn為CPn+p的全實(shí)子流形.

關(guān)于具有常數(shù)量曲率子流形,文獻(xiàn)[1]最早研究了空間形式中的常數(shù)量曲率超曲面,引入了一個(gè)自共軛的二階橢圓算子,這個(gè)算子現(xiàn)在仍是研究具有常數(shù)量曲率子流形的重要工具.文獻(xiàn)[2-4]應(yīng)用這個(gè)算子研究了各種空間中具有常數(shù)量曲率的子流形.本文試圖將文獻(xiàn)[1-2]的結(jié)果推廣到復(fù)射影空間,得到了:

定理 1 設(shè)Mn是CPn+p中具有常數(shù)量曲率和平行單位平均曲率向量場的緊致全實(shí)子流形,且其標(biāo)準(zhǔn)常數(shù)曲率R≥1,則

2 準(zhǔn)備工作

3 定理證明

[1]Cheng S Y,Yau S T.Hypersurfaceswith constant scalar curvature[J].M ath.Ann.,1977,225:195-204.

[2]Zhang J F.An rigidity theorem for submanifolds in Sn+pwith constant scalar curvature[J].Zhejiang Univ. Science:A,2005,6(4)1:322-328.

[3]宋衛(wèi)東,劉敏.關(guān)于局部對(duì)稱共形平坦空間中具有常數(shù)量曲率的子流形[J].數(shù)學(xué)物理學(xué)報(bào),2010,30:1102-1110.

[4]朱靜勇.Com p lete totally real pseudo umbilical subm anifolds with constant scalar curvature in a com p lex space form[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2011,27(1):116-122.

[5]張量.關(guān)于復(fù)射影空間的全實(shí)偽臍子流形[J].數(shù)學(xué)研究與評(píng)論,2008(2):421-428.

[6]Okumua M.Hypersurfaces and a pinching p roblem on the second fundamental tensor[J].Amer.J.Math., 1974,96:207-213.

[7]Li A M,Li J M.An intrinsic rigidity theorem for m inim al subm anifolds in sphere[J].A rch.M ath.,1992, 58:582-594.

On totally real subm anifolds with constant scalar carvatu re in the com p lex p ro jective space

Liu M in,Song Weidong

(College of M athem atics and Com puter Science,Anhui Norm al University,W uhu 241000,China)

In this paper,by choosing a suitable fram e field,we discussed the totally real subm anifolds with constant scalar curvture in the com p lex p rojective space and obtain a rigidity theorem that it becomes totally umbilical subm anifold and im p rove the related resu lts.

com p lex projective space,totally real subm anifolds,scalar curvature,totally umbilical

O186.12

A

1008-5513(2012)06-0749-08

2012-06-12.

安徽省高等學(xué)校優(yōu)秀青年人才基金(2011SQRL021ZD);安徽省高等學(xué)校自然科學(xué)研究項(xiàng)目基金(KJ2011Z149).

劉敏(1980-),碩士,講師,研究方向：子流形幾何.

2010 M SC:53C42,52B25