\(H\)-projectively equivalent four-dimensional Riemannian connections (Q1842482)

The authors consider two Kählerian metrics \(g\), \(\widehat {g}\) on a \(2n\)-dimensional manifold \(M\) with a complex structure \(J\). The Levi- Civita connections \(\nabla\) and \(\widehat {\nabla}\) are said to be \(H\)- projectively equivalent if \[ \widehat\Gamma_{j\phantom{i}k}^{\phantom{j}i\phantom{k}}- \Gamma_{j\phantom{i}k}^{\phantom{j}i\phantom{k}}= {2\delta_(}{_j^i \psi_{k)}} - 2\psi_l {J_(}{_j^l J^i_{k)}}. \] See \textit{N. S. Sinyukov}, Geodesic mappings of Riemannian space (Nauka, Moskva) (Russian) (1979; Zbl 0637.53020). In this case \(\psi_k = \partial _k \psi\), and the authors consider the 2-form field \(m_{ij} = \text{exp} (2\psi) g_{is} g_{jt} \widehat {g}^{st} - \lambda g_{ij}\). They prove that for \(n = 2\), \(|m_{ij}|\) has the Segre characteristic [(11)(11)] or [(22)], and with the use of the skew- normal frame technique they find all \(H\)-projectively equivalent Riemannian connections of four-dimensional Kählerian spaces.