An exploration of connections and curvature in the presence of singularities
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929607187365888 |
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| author | Herbig, Hans-Christian Esquivel, William Osnayder Clavijo |
| author_facet | Herbig, Hans-Christian Esquivel, William Osnayder Clavijo |
| contents | We develop the notions of connections and curvature for general Lie-Rinehart algebras without using smoothness assumptions on the base space. We present situations when a connection exists. E.g., this is the case when the underlying module is finitely generated. We show how the group of module automorphism acts as gauge transformations on the space of connections. When the underlying module is projective we define a version of the Chern character reproducing results of Hideki Ozeki. We discuss various examples of flat connections and the associated Maurer-Cartan equations. We provide examples of Levi-Civita connections on singular varieties and singular differential spaces with non-zero Riemannian curvature. The main observation is that for quotient singularities, even though the metric degenerates along strata, the poles of the Christoffel symbols are removable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04829 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An exploration of connections and curvature in the presence of singularities Herbig, Hans-Christian Esquivel, William Osnayder Clavijo Differential Geometry Mathematical Physics Commutative Algebra Algebraic Geometry primary 53C05, secondary 13C05, 13A50, 17B63, 13D02 We develop the notions of connections and curvature for general Lie-Rinehart algebras without using smoothness assumptions on the base space. We present situations when a connection exists. E.g., this is the case when the underlying module is finitely generated. We show how the group of module automorphism acts as gauge transformations on the space of connections. When the underlying module is projective we define a version of the Chern character reproducing results of Hideki Ozeki. We discuss various examples of flat connections and the associated Maurer-Cartan equations. We provide examples of Levi-Civita connections on singular varieties and singular differential spaces with non-zero Riemannian curvature. The main observation is that for quotient singularities, even though the metric degenerates along strata, the poles of the Christoffel symbols are removable. |
| title | An exploration of connections and curvature in the presence of singularities |
| topic | Differential Geometry Mathematical Physics Commutative Algebra Algebraic Geometry primary 53C05, secondary 13C05, 13A50, 17B63, 13D02 |
| url | https://arxiv.org/abs/2411.04829 |