Totally complex submanifolds in \(\mathbb{H} P^ m(1)\) (Q1900006)

The author proves pinching theorems for the square norm \(|\sigma|^2\) of the second fundamental form and for the sectional curvature \(K\) of compact totally complex immersed submanifolds \(M\) of a quaternionic projective space \(\mathbb{H} P^m (1)\). More precisely, if \(|\sigma|^2 \leq n\), where \(2n\) denotes the real dimension of \(M\), then either \(M\) is totally geodesic in \(\mathbb{H} P^m (1)\), or \(M\) is congruent to the standard embedding \(\mathbb{C} P^1 (1/2) \to \mathbb{C} P^2(1) \to \mathbb{H} P^2 (1)\), or \(M\) is congruent to the standard embedding \(Q^n \to \mathbb{C} P^{n + 1} (1) \to \mathbb{H} P^{n + 1} (1)\), \(n \geq 3\), of the complex quadric \(Q^n\). If \(n \geq 2\) and \(K \geq 1/8\), then \(M\) is either totally geodesic or congruent to the standard embedding \(\mathbb{C} P^n (1/2) \to \mathbb{C} P^{n(n + 3)/2} (1) \to \mathbb{H} P^{n(n+3)/2} (1)\). Both results are obtained by reducing the problem to a classification result by \textit{P. Coulton} and \textit{H. Gauchman} [Kodai Math. J. 12, No. 3, 296-307 (1989; Zbl 0691.53047)].