Geodesic
On Ricci curvature of totally real submanifolds in a quaternion projective space
Archivum mathematicum, Tome 38 (2002) no. 4, pp. 297-305 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
Classification : 53C26, 53C40, 53C42
Keywords: Ricci curvature; totally real submanifolds; quaternion projective space 
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     title = {On {Ricci} curvature of totally real submanifolds in a quaternion projective space},
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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/

[1] Chen B. Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60 (1993), 568–578. | MR | Zbl

[2] Chen B. Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension. Glasgow Math. J. 41 (1999), 33–41. | MR

[3] Chen B. Y.: On Ricci curvature of isotropic and Lagrangian submanifolds in the complex space forms. Arch. Math. 74 (2000), 154–160. | MR

[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

[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. | MR | Zbl

[6] Chen B. Y., Houh C. S.: Totally real submanifolds of a quaternion projective space. Ann. Mat. Pura Appl. 120 (1979), 185–199. | MR | Zbl

[7] Ishihara S.: Quaternion Kaehlerian manifolds. J. Differential Geom. 9 (1974), 483–500. | MR | Zbl