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 
@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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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/