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

Ximin Liu

Archivum Mathematicum (2002)

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

Abstract

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Let M n be a Riemannian n -manifold. Denote by S ( p ) and 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 Q P m ( c ) satisfies S ≤ ( ( n - 1 ) c + 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 Q P m ( c ) satisfies Ric ¯ = ( n - 1 ) c + n 2 4 H 2 identically, then it is minimal.

How to cite

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Liu, 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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  1. Chen B. Y., Some pinching and classification theorems for minimal submanifolds, Arch. Math. 60 (1993), 568–578. (1993) Zbl0811.53060MR1216703
  2. Chen B. Y., Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999), 33–41. (1999) MR1689730
  3. Chen B. Y., On Ricci curvature of isotropic and Lagrangian submanifolds in the complex space forms, Arch. Math. 74 (2000), 154–160. MR1735232
  4. Chen B. Y., Dillen F., Verstraelen L., Vrancken L., Totally real submanifolds of C P n satisfying a basic equality, Arch. Math. 63 (1994), 553–564. (1994) MR1300757
  5. Chen B. Y., Dillen F., Verstraelen L., Vrancken L., An exotic totally real minimal immersion of S 3 and C P 3 and its characterization, Proc. Royal Soc. Edinburgh, Sect. A, Math. 126 (1996), 153–165. (1996) Zbl0855.53011MR1378838
  6. Chen B. Y., Houh C. S., Totally real submanifolds of a quaternion projective space, Ann. Mat. Pura Appl. 120 (1979), 185–199. (1979) Zbl0413.53031MR0551066
  7. Ishihara S., Quaternion Kaehlerian manifolds, J. Differential Geom. 9 (1974), 483–500. (1974) Zbl0297.53014MR0348687

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