On local isometric immersions into complex and quaternionic projective spaces. (English) Zbl 1249.53079

Rivertz, Hans Jakob

Arch. Math., Brno 47, No. 4, 251-256 (2011).

The author studies local isometric immersions of complex projective spaces \(\mathbb {C}P^n \to \mathbb {C}P^m\) and of quaternionic projective spaces \(\mathbb {H}P^n \to \mathbb {H}P^m\), i.e.\(n<m\). Assume \(U \subset \mathbb {C}P^n\) and \(V \subset \mathbb {H}P^n\) are open and consider standard embeddings \(U \subset \mathbb {C}P^m\) and \(V \subset \mathbb {H}P^m\). It is shown that, for low codimensions, these embeddings are rigid in the class of real homothetic immersions. Assuming \(f\: U \to \mathbb {C}P^m\) and \(f'\: V \to \mathbb {H}P^m\) are (arbitrary) isometric immersions, it is shown that if \(m < (4/3)n-2/3\) then \(f\) is totally geodesic and if \(m < (4/3)n-5/6\) then \(f'\) is totally geodesic.

Reviewer: Josef Šilhan (Brno)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C24	Rigidity results
53C30	Differential geometry of homogeneous manifolds

Keywords:

submanifold; homogeneous space; symmetric space