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Totally real surfaces in \(QP^2\) with parallel mean curvature vector. (English) Zbl 0932.53019

Let \(M\) be a totally real surface with parallel mean curvature vector field in the quaternionic projective plane \(QP^2\), and denote by \(S\) the square of the length of the second fundamental form and by \(H\) the mean curvature of \(M\). The author proves the following results: (1) \(S\geq 3H^2\), and if equality holds then \(M\) has parallel second fundamental form; (2) \(3S\geq 8H^2\), and equality holds if and only if \(M\) is totally geodesic; (3) if (a) \(M\) has constant Gaussian curvature, or (b) \(M\) is compact and has nonnegative Gaussian curvature, or (c) \(M\) has parallel second fundamental form, then \(M\) is either totally geodesic or flat.

MSC:

53B25 Local submanifolds
53B35 Local differential geometry of Hermitian and Kählerian structures
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