Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic space
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913237353627648 |
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| author | Rungi, Nicholas |
| author_facet | Rungi, Nicholas |
| contents | For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise to a non-degenerate Riemannian metric on the smooth locus of the subset consisting of maximal representations inside the character variety. In the case $n=2$, we carefully study the properties of the Riemannian metric on the maximal connected components, proving that it is compatible with the orbifold structure and finding some totally geodesic sub-varieties. Then, in the general case, we explain when a representation with Zariski closure contained in $\mathrm{SO}_0(2,3)$ represents a smooth or orbifold point in the maximal $\mathrm{SO}_0(2,n+1)$-character variety and we show that the associated space is totally geodesic for any $n\ge 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_09351 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic space Rungi, Nicholas Differential Geometry Representation Theory For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise to a non-degenerate Riemannian metric on the smooth locus of the subset consisting of maximal representations inside the character variety. In the case $n=2$, we carefully study the properties of the Riemannian metric on the maximal connected components, proving that it is compatible with the orbifold structure and finding some totally geodesic sub-varieties. Then, in the general case, we explain when a representation with Zariski closure contained in $\mathrm{SO}_0(2,3)$ represents a smooth or orbifold point in the maximal $\mathrm{SO}_0(2,n+1)$-character variety and we show that the associated space is totally geodesic for any $n\ge 3$. |
| title | Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic space |
| topic | Differential Geometry Representation Theory |
| url | https://arxiv.org/abs/2309.09351 |