The type number on real hypersurfaces in a quaternion space form.

*(English)*Zbl 1034.53061From the author’s introduction: Let \(M_n(c)\) be a \(4n\)-dimensional quaternion space form with the metric \(g\) of constant quaternion sectional curvature \(8c\). Let \(M\) be a connected real hypersurface in \(M_n(c)\) with the induced metric. B.-Y. Chen and T. Nagano [Duke Math. J. 45, 405–425 (1978; Zbl 0384.53024)] investigated totally geodesic submanifolds in Riemannian symmetric spaces, and as one of their results the following holds

Theorem A. Spheres and hyperbolic spaces are only simply connected irreducible symmetric spaces admitting a totally geodesic hypersurface.

It will be an interesting problem to study the type number \(t\) of real hypersurfaces in simply connected irreducible symmetric spaces excepted for spheres and hyperbolic spaces.

As a partial answer, it is known that there exists a point such that \(t(p)\geq 2\) in any real hypersurface in complex space forms with nonzero constant holomorphic sectional curvature and complex dimension \(\geq 3\). Naturally we can consider the following question. Are similar facts true for \(M_\mu(c)\)? We answer this question affirmatively, i.e., we prove the following main theorem. Let \(M\) be a connected real hypersurface in \(M_n(c)\) \((c\neq 0, n\geq 2)\). Then there exists a point \(p\) in \(M\) such that \(t(p)\geq 2\).

Theorem A. Spheres and hyperbolic spaces are only simply connected irreducible symmetric spaces admitting a totally geodesic hypersurface.

It will be an interesting problem to study the type number \(t\) of real hypersurfaces in simply connected irreducible symmetric spaces excepted for spheres and hyperbolic spaces.

As a partial answer, it is known that there exists a point such that \(t(p)\geq 2\) in any real hypersurface in complex space forms with nonzero constant holomorphic sectional curvature and complex dimension \(\geq 3\). Naturally we can consider the following question. Are similar facts true for \(M_\mu(c)\)? We answer this question affirmatively, i.e., we prove the following main theorem. Let \(M\) be a connected real hypersurface in \(M_n(c)\) \((c\neq 0, n\geq 2)\). Then there exists a point \(p\) in \(M\) such that \(t(p)\geq 2\).