ON CURVATURE PINCHING FOR TOTALLY REAL SUBMANIFOLDS OF $H^n$(c)

Let S be the Ricci curvature of an n-dimensional compact minimal totally real submanifold M of a quaternion projective space $HP^n (c)$ of quaternion sectional curvature c. We proved that if $S \leq \frac{16}{3(n -2)}c$, then either $S \equiv \frac{4}{n - 1}c$ (i.e. M is totally geodesic or $S \equiv \frac{16}{3(n - 2)}c$. All compact minimal totally real submanifolds of $HP^n (c)$ satisfy in $S \equiv \frac{16}{3(n - 2)}c$ are determined.

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