On isometric immersions into complex space forms. (English) Zbl 0806.53019

The authors consider the question of whether an isometric immersion of a connected Kähler manifold into a non-flat complex space form is holomorphic. The main result shows that this is so if the index of relative nullity is positive everywhere. This result has many applications. For example, if the codimension is sufficiently low and the sectional curvatures of the submanifold at one point are at most equal to those of the ambient complex space form, then the immersion must be holomorphic. Other applications are of a global nature or relate to the problem of reducing the codimension.


53B25 Local submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B35 Local differential geometry of Hermitian and Kählerian structures
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