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Jacobi vector fields and geodesic tubes in certain Kähler manifolds. (English) Zbl 0979.53062
Slovák, Jan (ed.) et al., The proceedings of the 19th Winter School “Geometry and physics”, Srní, Czech Republic, January 9-15, 1999. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 63, 43-52 (2000).
The authors give a characterization of Kähler manifolds with constant holomorphic sectional curvature in every real dimension \(2n\geq 4\) using properties of the shape operator. If \(\left( M,g,J\right) \) is a connected Kähler manifold with constant holomorphic sectional curvature, then \(M\) has the property that sufficiently small tubes about an arbitrary geodesic (geodesic tubular hypersurfaces) are quasi-umbilical hypersurfaces of \(M\), and conversely. The result is obtained from calculating the shape operator on a tube of given radius about a geodesic and looking at the multiplicities of the eigenvalues and properties of the corresponding eigenspaces of the operator.
For the entire collection see [Zbl 0940.00040].
53C40 Global submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C20 Global Riemannian geometry, including pinching