Projectively equivalent para-Kaehler and para-Kaehler-Einstein metrics with non-parallel Benenti tensors and their normal forms in dimension four

The study of projectively equivalent metrics, i.e., metrics sharing the same unparametrized geodesics, is a classical and well-established area of investigation. In the Kaehler context, such branch of research goes by the name of c-projective geometry: it mainly studies c-projectively equivalent metrics, i.e., Kaehler metrics sharing the same curves that are the complex analogue of the geodesics, called J-planar, where J is the complex structure. In this paper, we develop the theory of the projective equivalence in the para-Kaehler context by studying para-Kaehler metrics sharing the same T-planar curves, where T is the para-complex structure: we call such metrics pc-projectively equivalent. After establishing some general results in arbitrary dimension, we focus on the 4-dimensional case. One of the main achievement is a local description of 4-dimensional pc-projectively (but not affinely) equivalent metrics and, as an application of this result, we characterize which of them are of Einstein type.