# On Ricci curvature of totally real submanifolds in a quaternion projective space

Archivum Mathematicum (2002)

- Volume: 038, Issue: 4, page 297-305
- ISSN: 0044-8753

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topLiu, Ximin. "On Ricci curvature of totally real submanifolds in a quaternion projective space." Archivum Mathematicum 038.4 (2002): 297-305. <http://eudml.org/doc/248946>.

@article{Liu2002,

abstract = {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.},

author = {Liu, Ximin},

journal = {Archivum Mathematicum},

keywords = {Ricci curvature; totally real submanifolds; quaternion projective space; Ricci curvature; totally real submanifolds; quaternion projective space},

language = {eng},

number = {4},

pages = {297-305},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {On Ricci curvature of totally real submanifolds in a quaternion projective space},

url = {http://eudml.org/doc/248946},

volume = {038},

year = {2002},

}

TY - JOUR

AU - Liu, Ximin

TI - On Ricci curvature of totally real submanifolds in a quaternion projective space

JO - Archivum Mathematicum

PY - 2002

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 038

IS - 4

SP - 297

EP - 305

AB - 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.

LA - eng

KW - Ricci curvature; totally real submanifolds; quaternion projective space; Ricci curvature; totally real submanifolds; quaternion projective space

UR - http://eudml.org/doc/248946

ER -

## References

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