On a Kaehler hypersurface of a complex space form with the recurrent second fundamental form
On a Kaehler hypersurface of a complex space form with the recurrent second fundamental form
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具有循环第二基本形式的复杂空间形式的凯勒超曲面
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Y. Matsuyama
中科院分区：
文献类型：
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作者：
Yumetaro Mashiko;Satoshi Kurosu;Y. Matsuyama
抽象。本文的目的是证明：设M是复空间形式M~（c）的Kaehler超曲面，其全纯截面曲率为c。则下列条件是等价的：（A）M具有递归的第二基本形式。(B)M具有双流第二基本形式。(C)导子R（X;Y）与第二基本形式是可交换的。(D)M在M~（c）. M.S.C.中是全测地的。2000：53 C40，53 B25.关键词：Kaehler超曲面，递归第二基本形式，全测地线. 1引言设M~（c）是复维数n +1且具有常全纯截面曲率c的复空间形式（即完备的单连通Kaehler流形），若c为正，则M~（c）是复射影空间Pn +1（c）且具有常全纯截面曲率c的Fubini-Study度量。若c为负，则M~（c）是复双曲空间Dn +1（c），其Bergman度量具有常纯截面曲率
Abstract. The purpose of the present paper is to prove: Let M be aKaehler hypersurface of a complex space form M~ ( c ) of constant holomor-phic sectional curvature c . Then the following conditions are equivalent:(A) M has the recurrent second fundamental form. (B) M has the bire-current second fundamental form. (C) the derivation R ( X;Y ) and thesecond fundamental form are commutative. (D) M is totally geodesic in M~ ( c ).M.S.C. 2000: 53C40, 53B25.Key words: Kaehler hypersurface, the recurrent second fundamental form, totallygeodesic. 1 Introduction Let M~ ( c ) be a complex space form (i.e. a complete, simply-connected Kaehler mani-fold) of the complex dimension n +1 with constant holomorphic sectional curvature c .If c is positive, then M~ ( c ) is the complex projective space P n +1 ( c ) with the Fubini-Study metric of constant holomorphic sectional curvature c . If c is negative, then M~ ( c ) is the complex hyperbolic space D n +1 ( c ) with the Bergman metric of constantholomorphic sectional curvature