Let $M^n$ be a Riemannian $n$-manifold. Denote by $S(p)$ and $\overline{\operatorname{Ric}}(p)$ the Ricci tensor and the maximum Ricci curvature on $M^n$, respectively. In this paper we prove that every totally real submanifolds of a quaternion projective space $QP^m(c)$ satisfies $S\le ((n-1)c+\frac{n^2}{4}H^2)g$, where $H^2$ and $g$ are the square mean curvature function and metric tensor on $M^n$, respectively. The equality holds identically if and only if either $M^n$ is totally geodesic submanifold or $n=2$ and $M^n$ is totally umbilical submanifold. Also we show that if a Lagrangian submanifold of $QP^m(c)$ satisfies $\overline{\operatorname{Ric}}=(n-1)c+\frac{n^2}{4}H^2$ identically, then it is minimal.
Let $M^n$ be a Riemannian $n$-manifold. Denote by $S(p)$ and $\overline{\operatorname{Ric}}(p)$ the Ricci tensor and the maximum Ricci curvature on $M^n$, respectively. In this paper we prove that every totally real submanifolds of a quaternion projective space $QP^m(c)$ satisfies $S\le ((n-1)c+\frac{n^2}{4}H^2)g$, where $H^2$ and $g$ are the square mean curvature function and metric tensor on $M^n$, respectively. The equality holds identically if and only if either $M^n$ is totally geodesic submanifold or $n=2$ and $M^n$ is totally umbilical submanifold. Also we show that if a Lagrangian submanifold of $QP^m(c)$ satisfies $\overline{\operatorname{Ric}}=(n-1)c+\frac{n^2}{4}H^2$ identically, then it is minimal.
@article{ARM_2002_38_4_a5,
author = {Liu, Ximin},
title = {On {Ricci} curvature of totally real submanifolds in a quaternion projective space},
journal = {Archivum mathematicum},
pages = {297--305},
year = {2002},
volume = {38},
number = {4},
mrnumber = {1942659},
zbl = {1090.53052},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ARM_2002_38_4_a5/}
}
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AU - Liu, Ximin
TI - On Ricci curvature of totally real submanifolds in a quaternion projective space
JO - Archivum mathematicum
PY - 2002
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EP - 305
VL - 38
IS - 4
UR - http://geodesic.mathdoc.fr/item/ARM_2002_38_4_a5/
LA - en
ID - ARM_2002_38_4_a5
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%0 Journal Article
%A Liu, Ximin
%T On Ricci curvature of totally real submanifolds in a quaternion projective space
%J Archivum mathematicum
%D 2002
%P 297-305
%V 38
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%U http://geodesic.mathdoc.fr/item/ARM_2002_38_4_a5/
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Liu, Ximin. On Ricci curvature of totally real submanifolds in a quaternion projective space. Archivum mathematicum, Tome 38 (2002) no. 4, pp. 297-305. http://geodesic.mathdoc.fr/item/ARM_2002_38_4_a5/
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