| Title:
|
On Ricci curvature of totally real submanifolds in a quaternion projective space (English) |
| Author:
|
Liu, Ximin |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
38 |
| Issue:
|
4 |
| Year:
|
2002 |
| Pages:
|
297-305 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Ricci curvature |
| Keyword:
|
totally real submanifolds |
| Keyword:
|
quaternion projective space |
| MSC:
|
53C26 |
| MSC:
|
53C40 |
| MSC:
|
53C42 |
| idZBL:
|
Zbl 1090.53052 |
| idMR:
|
MR1942659 |
| . |
| Date available:
|
2008-06-06T22:31:03Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107843 |
| . |
| Reference:
|
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| Reference:
|
[2] Chen B. Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension.Glasgow Math. J. 41 (1999), 33–41. MR 1689730 |
| Reference:
|
[3] Chen B. Y.: On Ricci curvature of isotropic and Lagrangian submanifolds in the complex space forms.Arch. Math. 74 (2000), 154–160. MR 1735232 |
| Reference:
|
[4] Chen B. Y., Dillen F., Verstraelen L., Vrancken L.: Totally real submanifolds of $CP^n$ satisfying a basic equality.Arch. Math. 63 (1994), 553–564. MR 1300757 |
| Reference:
|
[5] Chen B. Y., Dillen F., Verstraelen L., Vrancken L.: An exotic totally real minimal immersion of $S^3$ and $CP^3$ and its characterization.Proc. Royal Soc. Edinburgh, Sect. A, Math. 126 (1996), 153–165. Zbl 0855.53011, MR 1378838 |
| Reference:
|
[6] Chen B. Y., Houh C. S.: Totally real submanifolds of a quaternion projective space.Ann. Mat. Pura Appl. 120 (1979), 185–199. Zbl 0413.53031, MR 0551066 |
| Reference:
|
[7] Ishihara S.: Quaternion Kaehlerian manifolds.J. Differential Geom. 9 (1974), 483–500. Zbl 0297.53014, MR 0348687 |
| . |
