Quasi-constant holomorphic sectional curvatures of tangent bundles with general natural Kähler structures (Q2919606)

A Kähler manifold \((M,g,J)\), endowed with a unit vector field \(\xi\), is said to be of quasi-constant holomorphic sectional curvature if for any holomorphic section span\(\{X,JX\}\), generated by the unit vector \(X\in T_pM,\;p\in M\), the sectional curvature \(R(X,JX,JX,X)\) depends only on the point \(p\) and on the angle between the holomorphic plane and the unit vector field \(X\).NEWLINENEWLINE A characterization of the Kähler manifolds of quasi-constant holomorphic sectional curvature, given by \textit{C.--L. Bejan} and \textit{M. Benyounes} in [J. Geom. 88, No. 1--2, 1--14 (2008; Zbl 1172.53047)], involves a special expression of the curvature tensor field. By using this characterization, the author proves that the general natural Kählerian tangent bundles \((TM,G,J)\) of quasi-constant holomorphic curvature are only those of constant holomorphic sectional curvature, classified in [\textit{S. Druţă} and \textit{V. Oproiu}, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 53, No. 1, 149--166 (2007; Zbl 1174.53040)]. This result is a generalization of Shur's theorem: if the sectional curvature of any holomorphic plane generated by the unit nonzero vector field \(X\in T_pM\) depends only on the angle with the Liouville vector field and on the point \(p\in M\), then \(TM\) has constant holomorphic sectional curvature.