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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.07586 |
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| _version_ | 1866912319561269248 |
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| author | Draper, Cristina Martín-González, Cándido |
| author_facet | Draper, Cristina Martín-González, Cándido |
| contents | We provide an independent proof of the classification of the maximal totally geodesic submanifolds of the symmetric spaces $G_2$ and $G_2/SO(4)$, jointly with very natural descriptions of all of these submanifolds. The description of the totally geodesic submanifolds of $G_2$ is in terms of (1) principal subalgebras of $\mathfrak{g}_2$; (2) stabilizers of nonzero points of $\mathbb{R}^7$; (3) stabilizers of associative subalgebras; (4) the set of order two elements in $G_2$ (and its translations). The space $G_2/SO(4)$ is identified with the set of associative subalgebras of $\mathbb{R}^7$ and its maximal totally geodesic submanifolds can be described as the associative subalgebras adapted to a fixed principal subalgebra, the associative subalgebras orthogonal to a fixed nonzero vector, the associative subalgebras containing a fixed nonzero vector, and the associative subalgebras intersecting both a fixed associative subalgebra and its orthogonal. A second description is included in terms of Grassmannians, the advantage of which is that the associated Lie triple systems are easily described in matrix form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A perspective on totally geodesic submanifolds of the symmetric space $G_2/SO(4)$ Draper, Cristina Martín-González, Cándido Differential Geometry We provide an independent proof of the classification of the maximal totally geodesic submanifolds of the symmetric spaces $G_2$ and $G_2/SO(4)$, jointly with very natural descriptions of all of these submanifolds. The description of the totally geodesic submanifolds of $G_2$ is in terms of (1) principal subalgebras of $\mathfrak{g}_2$; (2) stabilizers of nonzero points of $\mathbb{R}^7$; (3) stabilizers of associative subalgebras; (4) the set of order two elements in $G_2$ (and its translations). The space $G_2/SO(4)$ is identified with the set of associative subalgebras of $\mathbb{R}^7$ and its maximal totally geodesic submanifolds can be described as the associative subalgebras adapted to a fixed principal subalgebra, the associative subalgebras orthogonal to a fixed nonzero vector, the associative subalgebras containing a fixed nonzero vector, and the associative subalgebras intersecting both a fixed associative subalgebra and its orthogonal. A second description is included in terms of Grassmannians, the advantage of which is that the associated Lie triple systems are easily described in matrix form. |
| title | A perspective on totally geodesic submanifolds of the symmetric space $G_2/SO(4)$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2504.07586 |