Minimal submanifolds inS N andR N

On totally real minimal submanifolds in $CP^{n}(c)$

“…One easily checks that $\langle, \rangle$ is a positive definite inner product on $N_{1}(x)$ . The following lemmas can be proved in the case of totally real, minimal submanifolds in $CP^{n}(c)( [4])$ : LEMMA 1. Let $M^{n}$ be a totally real, minimal submanifolds in $CP^{n}(c)$ .…”

“…Recently, in [4] we proved : Let $M^{n}$ be a minimal $n(\geq 4)$ -submanifold in a Euclidean N-sphere $S^{N}$ of radius 1 (resp. a Euclidean N-space $R^{N}$ ) which has at most two principal curvatures in the direction of any normal which satisfy that if exactly two are distinct, then we assume those multiplicites $\geq 2$ .…”

On a 2-Dimensional Einstein Kaehler Submanifold of a Complex Space Form

“…[4]), where -y is the eigenvalue of A1. Hence, we obtain 22 all* /2 3a22* Since (all a22 all* a220) 0 0, we have a/3 = 0.…”

On a 2-dimensional Einstein Kaehler submanifold of a complex space form

“…Using the minimal polynomial of Ax, we can show that {X\AXX = yX) is differentiable (e.g. [4]), where y is the eigenvalue of Ax. Thus we may assume that Ax is diagonalized with respect to {Ex, E2, Ex,, E2»} in U.…”