The canonical generalised Levi-Civita connection and its curvature
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arXiv
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| Format: | Preprint |
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2025
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| author | Cortés, Vicente Mackevicius, Matas Mohaupt, Thomas Schiller, Oskar |
| author_facet | Cortés, Vicente Mackevicius, Matas Mohaupt, Thomas Schiller, Oskar |
| contents | Given a (semi-Riemannian) generalised metric $\mathcal G$ and a divergence operator $\mathrm{div}$ on an exact Courant algebroid $E$, we geometrically construct a canonical generalised Levi-Civita connection $D^{\mathcal G, \mathrm{div}}$ for these data. In this way we provide a resolution of the problem of non-uniqueness of generalised Levi-Civita connections. Since the generalised Riemann tensor of $D^{\mathcal G, \mathrm{div}}$ is an invariant of the pair $(\mathcal G, \mathrm{div})$, we no longer need to discard curvature components which depend on the choice of the generalised connection. As a main result we decompose the generalised Riemann curvature tensor of $D^{\mathcal G, \mathrm{div}}$ in terms of classical (non-generalised) geometric data. Based on this set of master formulas we derive a comprehensive curvature tool-kit for applications in generalised geometry. This includes decompositions for the full generalised Ricci tensor, the generalised Ricci tensor, and three generalised scalar-valued curvature invariants, two of which are new. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The canonical generalised Levi-Civita connection and its curvature Cortés, Vicente Mackevicius, Matas Mohaupt, Thomas Schiller, Oskar Differential Geometry General Relativity and Quantum Cosmology High Energy Physics - Theory 53D18 (Generalized geometry a la Hitchin), 83C10 (Equations of motion in general relativity and gravitational theory) Given a (semi-Riemannian) generalised metric $\mathcal G$ and a divergence operator $\mathrm{div}$ on an exact Courant algebroid $E$, we geometrically construct a canonical generalised Levi-Civita connection $D^{\mathcal G, \mathrm{div}}$ for these data. In this way we provide a resolution of the problem of non-uniqueness of generalised Levi-Civita connections. Since the generalised Riemann tensor of $D^{\mathcal G, \mathrm{div}}$ is an invariant of the pair $(\mathcal G, \mathrm{div})$, we no longer need to discard curvature components which depend on the choice of the generalised connection. As a main result we decompose the generalised Riemann curvature tensor of $D^{\mathcal G, \mathrm{div}}$ in terms of classical (non-generalised) geometric data. Based on this set of master formulas we derive a comprehensive curvature tool-kit for applications in generalised geometry. This includes decompositions for the full generalised Ricci tensor, the generalised Ricci tensor, and three generalised scalar-valued curvature invariants, two of which are new. |
| title | The canonical generalised Levi-Civita connection and its curvature |
| topic | Differential Geometry General Relativity and Quantum Cosmology High Energy Physics - Theory 53D18 (Generalized geometry a la Hitchin), 83C10 (Equations of motion in general relativity and gravitational theory) |
| url | https://arxiv.org/abs/2507.17604 |