Hybrid connections on Hessian manifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915932430925824 |
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| author | Chéritat, Arnaud Tahar, Guillaume |
| author_facet | Chéritat, Arnaud Tahar, Guillaume |
| contents | A Hessian manifold $(M,D,g)$ is a manifold $M$ with a flat connection $D$ and a Riemannian or pseudo-Riemannian metric $g$ that is locally of the form $D^2 f$ for some function $f$. On a Hessian manifold $(M,D,g)$, we define a hybrid connection as an incompressible affine connection $\nabla$ that is projectively flat relative to $D$ (its unparametrized geodesics are aligned with the affine structure of $D$) and whose first-order infinitesimal holonomy at each point of $M$ is an infinitesimal isometry of the pseudo-Riemannian metric $g$. In this paper, we investigate the properties of hybrid connections, proving in particular that for a hybrid connection $\nabla$, the difference $\nabla-D$ is determined by the logarithmic differential of a function that serves as a Hessian potential for $g$. In the special case of pseudo-Euclidean manifolds, we identify canonical models and obtain in particular a new natural connection on the open unit ball that provides a compromise between properties of Cayley-Klein and Poincaré hyperbolic models. We also find a unique (up to a scaling) pseudo-Riemannian metric $h$ such that unparameterized geodesics of $\nabla$ have a constant speed with respect to the so-called isochrone metric $h$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12543 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hybrid connections on Hessian manifolds Chéritat, Arnaud Tahar, Guillaume Differential Geometry 53B05 (Primary), 53C05, 53C22, 53C25 (Secondary) A Hessian manifold $(M,D,g)$ is a manifold $M$ with a flat connection $D$ and a Riemannian or pseudo-Riemannian metric $g$ that is locally of the form $D^2 f$ for some function $f$. On a Hessian manifold $(M,D,g)$, we define a hybrid connection as an incompressible affine connection $\nabla$ that is projectively flat relative to $D$ (its unparametrized geodesics are aligned with the affine structure of $D$) and whose first-order infinitesimal holonomy at each point of $M$ is an infinitesimal isometry of the pseudo-Riemannian metric $g$. In this paper, we investigate the properties of hybrid connections, proving in particular that for a hybrid connection $\nabla$, the difference $\nabla-D$ is determined by the logarithmic differential of a function that serves as a Hessian potential for $g$. In the special case of pseudo-Euclidean manifolds, we identify canonical models and obtain in particular a new natural connection on the open unit ball that provides a compromise between properties of Cayley-Klein and Poincaré hyperbolic models. We also find a unique (up to a scaling) pseudo-Riemannian metric $h$ such that unparameterized geodesics of $\nabla$ have a constant speed with respect to the so-called isochrone metric $h$. |
| title | Hybrid connections on Hessian manifolds |
| topic | Differential Geometry 53B05 (Primary), 53C05, 53C22, 53C25 (Secondary) |
| url | https://arxiv.org/abs/2302.12543 |