Generalized Kähler Spaces in Eisenhart’s Sense Admitting a Holomorphically Projective Mapping

  • Miloš Z. Petrović
  • Ljubica S. Velimirović


We relax the conditions \(\underset{1}{\nabla }F=0\) and \(\underset{2}{\nabla }F=0\) with respect to the non-symmetric linear connections \(\underset{1}{\nabla }\) and \(\underset{2}{\nabla }\) and an almost complex structure F of a generalized Kähler space. In such a way we introduce a wider class of generalized Kähler spaces which admits a holomorphically projective mapping. These generalized Kähler spaces are named generalized Kähler spaces in Eisenhart’s sense, since they are defined as a particular case of Eisenhart’s generalized Riemannian spaces. Curvature tensors of generalized Kähler spaces in Eisenhart’s sense have some interesting properties that have been pointed out in the present paper. Also, we consider equitorsion holomorphically projective mappings and examine some invariant geometric objects with respect to these mappings. Our results are deduced without restrictive conditions that were given in some of the previous papers.


Generalized Riemannian space generalized Kähler space holomorphically projective mapping equitorsion mapping invariant geometric object 

Mathematics Subject Classification

Primary 53B05 Secondary 53B20 53B35 



This work was supported by Grant No. 174012 of the Ministry of Education, Science and Technological Development, Republic of Serbia. The authors are grateful to the anonymous referee whose suggestions led to construction of Example 3.1 and to conclusion that in real dimension \(n=2\) there does not exist any example of a generalized Kähler space in Eisenhart’s sense.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Miloš Z. Petrović
    • 1
  • Ljubica S. Velimirović
    • 1
  1. 1.Department of MathematicsFaculty of Sciences and MathematicsNišSerbia

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