Second order Einstein deformations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916438954999808 |
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| author | Nagy, Paul-Andi Semmelmann, Uwe |
| author_facet | Nagy, Paul-Andi Semmelmann, Uwe |
| contents | We study the integrability to second order of infinitesimal Einstein deformations on compact Riemannian and in particular on Kähler manifolds. We find a new way of expressing the necessary and sufficient condition for integrability to second order, which also gives a very clear and compact way of writing the Koiso obstruction. As an application we consider the Kähler case, where the condition can be further simplified and in complex dimension $3$ turns out to be purely algebraic. One of our main results is the complete and explicit description of infinitesimal Einstein deformation integrable to second order on the complex $2$-plane Grassmannian, which also has a quaternion Kähler structure. As a striking consequence we find that the symmetric Einstein metric on the Grassmannian $ \mathrm{Gr}_2(\bbC^{n+2})$ for $n$ odd is rigid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07391 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Second order Einstein deformations Nagy, Paul-Andi Semmelmann, Uwe Differential Geometry 32Q20, 53C26, 53C35, 53C15 We study the integrability to second order of infinitesimal Einstein deformations on compact Riemannian and in particular on Kähler manifolds. We find a new way of expressing the necessary and sufficient condition for integrability to second order, which also gives a very clear and compact way of writing the Koiso obstruction. As an application we consider the Kähler case, where the condition can be further simplified and in complex dimension $3$ turns out to be purely algebraic. One of our main results is the complete and explicit description of infinitesimal Einstein deformation integrable to second order on the complex $2$-plane Grassmannian, which also has a quaternion Kähler structure. As a striking consequence we find that the symmetric Einstein metric on the Grassmannian $ \mathrm{Gr}_2(\bbC^{n+2})$ for $n$ odd is rigid. |
| title | Second order Einstein deformations |
| topic | Differential Geometry 32Q20, 53C26, 53C35, 53C15 |
| url | https://arxiv.org/abs/2305.07391 |