A characterization of totally η -umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

Mayuko Kon

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1279-1287
  • ISSN: 0011-4642

Abstract

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We give a characterization of totally η -umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M n ( c ) , c ≠ 0 , n ≥ 3 , satisfies g ( A X , Y ) = a g ( X , Y ) for any X , Y ∈ T 0 ( x ) , a being a function, where T 0 is the holomorphic distribution on M , then M is a totally η -umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η -umbilical real hypersurfaces.

How to cite

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Kon, Mayuko. "A characterization of totally $\eta $-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form." Czechoslovak Mathematical Journal 58.4 (2008): 1279-1287. <http://eudml.org/doc/37903>.

@article{Kon2008,
abstract = {We give a characterization of totally $\eta $-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator $A$ of a real hypersurface $M$ of a complex space form $M^n(c)$, $c\ne 0$, $n\ge 3$, satisfies $g(AX,Y)=ag(X,Y)$ for any $X,Y\in T_0(x)$, $a$ being a function, where $T_0$ is the holomorphic distribution on $M$, then $M$ is a totally $\eta $-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of $\eta $-umbilical real hypersurfaces.},
author = {Kon, Mayuko},
journal = {Czechoslovak Mathematical Journal},
keywords = {real hypersurface; totally $\eta $-umbilical real hypersurface; ruled real hypersurface; real hypersurface; totally -umbilical real hypersurface; ruled real hypersurface},
language = {eng},
number = {4},
pages = {1279-1287},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A characterization of totally $\eta $-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form},
url = {http://eudml.org/doc/37903},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Kon, Mayuko
TI - A characterization of totally $\eta $-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1279
EP - 1287
AB - We give a characterization of totally $\eta $-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator $A$ of a real hypersurface $M$ of a complex space form $M^n(c)$, $c\ne 0$, $n\ge 3$, satisfies $g(AX,Y)=ag(X,Y)$ for any $X,Y\in T_0(x)$, $a$ being a function, where $T_0$ is the holomorphic distribution on $M$, then $M$ is a totally $\eta $-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of $\eta $-umbilical real hypersurfaces.
LA - eng
KW - real hypersurface; totally $\eta $-umbilical real hypersurface; ruled real hypersurface; real hypersurface; totally -umbilical real hypersurface; ruled real hypersurface
UR - http://eudml.org/doc/37903
ER -

References

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