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

Dajczer, Marcos; Rodríguez, Lucio

Math. Ann. 299, No. 2, 223-230 (1994).

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.

Reviewer: M.Dajczer (Rio de Janeiro)

MSC:

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

Keywords:

holomorphic immersion; sectional curvatures

References:

[1]	Abe, K.: Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions. T?hoku Math. J.25, 425-444 (1973) · Zbl 0283.53045
[2]	Burns, D., Burstall, F., de Bartolomeis, P., Rawnsley, J.: Stability of harmonic maps of K?hler manifolds. J. Diff. Geom.30, 579-594 (1989) · Zbl 0678.53062
[3]	Carlson, J., Toledo, D.: Harmonic mappings of K?hler manifolds to locally symmetric spaces. Publ. Math. Inst. Hautes ?tud. Sci.69, 173-201 (1989) · Zbl 0695.58010
[4]	Cecil, T.: Geometric applications of critical point theory to submanifold of complex projective space. Nagoya Math. J.55, 5-31 (1974) · Zbl 0291.53014
[5]	Dajczer, M., Rodr?guez, L.: Complete real K?hler submanifolds. J. Reine Angew. Math.419, 1-8 (1991) · Zbl 0726.53041
[6]	Dajczer, M., Tojeiro, R.: Submanifolds with nonparallel first normal bundle. (To appear in Can. Math. J.) · Zbl 0812.53015
[7]	Florit, L.: On submanifolds with nonpositive extrinsic curvature. Math. Ann.298, 187-192 (1994) · Zbl 0810.53011
[8]	Rodr?guez, L., Tribuzy, R.: Reduction of codimension of regular immersions. Math. Z.185, 321-331 (1984) · Zbl 0545.53044
[9]	Ryan, P.: K?hler manifolds as real hypersurfaces. Duke Math. J.40, 207-214 (1973) · Zbl 0257.53055
[10]	Siu, S., Yau, S.-T.: Compact K?hler manifolds of positive bisectional curvature. Invent. Math.59, 189-204 (1980) · Zbl 0442.53056