On the quaternionic sectional curvature of an indefinite quaternionic Kähler manifold
In this paper, the authors characterize indefinite quaternionic space forms by the condition: \(R(U, \varphi U,U, \varphi U)=0\) for all null vectors \(U\) and \(\varphi= I,J, K\) any local basis of the bundle of almost complex structures. By a completely different approach, the authors streighten out a proof given in [\textit{J. D. Pérez} and \textit{F. Santos}, Ann. Mat. Pura Appl., IV. Ser. 132, 383-398 (1982; Zbl 0513.53043)]. Notice that \(R(U,JU, U, JU) =0\) for all null vectors \(U\) is not sufficient for the constancy of the holomorphic sectional curvature of an indefinite Kähler manifold [\textit{A. Bonome}, \textit{R. Castro}, \textit{E. García-Río} and \textit{L. M. Hervella} [C. R. Acad. Sci., Paris, Sér. I 315, No. 11, 1183-1187 (1992; Zbl 0765.53051)]. This makes a substantial difference with respect to the complex case. Lastly, using the expression of the Jacobi operator along null geodesics, the authors show the nonexistence of conjugate points along null geodesics in indefinite quaternionic space forms.
scientific article; zbMATH DE number 895937
scientific article