Projectively equivalent para-Kaehler and para-Kaehler-Einstein metrics with non-parallel Benenti tensors and their normal forms in dimension four
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908748334759936 |
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| author | Manno, Gianni Salis, Filippo |
| author_facet | Manno, Gianni Salis, Filippo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02159 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Projectively equivalent para-Kaehler and para-Kaehler-Einstein metrics with non-parallel Benenti tensors and their normal forms in dimension four Manno, Gianni Salis, Filippo Differential Geometry 53A20, 53C15, 53C25, 32Q15, 32Q20 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. |
| title | Projectively equivalent para-Kaehler and para-Kaehler-Einstein metrics with non-parallel Benenti tensors and their normal forms in dimension four |
| topic | Differential Geometry 53A20, 53C15, 53C25, 32Q15, 32Q20 |
| url | https://arxiv.org/abs/2601.02159 |