A family of left-invariant SKT metrics on the exceptional Lie group $G_2$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915242269736960 |
|---|---|
| author | Pham, David N. |
| author_facet | Pham, David N. |
| contents | For a complex manifold $(M,J)$, an SKT (or pluriclosed) metric is a $J$-Hermitian metric $g$ whose fundamental form $ω:=g(J\cdot,\cdot)$ satisfies the condition $\partial\overline{\partial}ω=0$. As such, an SKT metric can be regarded as a natural generalization of a Kähler metric. In this paper, the exceptional Lie group $G_2$ is equipped with a left-invariant integrable almost complex structure $\mathcal{J}$ via the Samelson construction and a 7-parameter family of $\mathcal{J}$-Hermitian metrics is constructed. From this 7-parameter family, the members which are SKT are calculated. The result is a 3-parameter family of left-invariant SKT metrics on $G_2$. As a special case, the aforementioned family of SKT metrics contains all bi-invariant metrics on $G_2$. In addition, this 3-parameter family of left-invariant SKT metrics are also invariant under the right action of a certain maximal torus $T$ of $G_2$. Conversely, it is shown that if $g$ is a left-invariant $\mathcal{J}$-Hermitian metric on $G_2$ such that $g$ is invariant under the right action of $T$ and for which $(g,\mathcal{J})$ is SKT, then $g$ must belong to this 3-parameter family of left-invariant SKT metrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10037 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A family of left-invariant SKT metrics on the exceptional Lie group $G_2$ Pham, David N. Differential Geometry For a complex manifold $(M,J)$, an SKT (or pluriclosed) metric is a $J$-Hermitian metric $g$ whose fundamental form $ω:=g(J\cdot,\cdot)$ satisfies the condition $\partial\overline{\partial}ω=0$. As such, an SKT metric can be regarded as a natural generalization of a Kähler metric. In this paper, the exceptional Lie group $G_2$ is equipped with a left-invariant integrable almost complex structure $\mathcal{J}$ via the Samelson construction and a 7-parameter family of $\mathcal{J}$-Hermitian metrics is constructed. From this 7-parameter family, the members which are SKT are calculated. The result is a 3-parameter family of left-invariant SKT metrics on $G_2$. As a special case, the aforementioned family of SKT metrics contains all bi-invariant metrics on $G_2$. In addition, this 3-parameter family of left-invariant SKT metrics are also invariant under the right action of a certain maximal torus $T$ of $G_2$. Conversely, it is shown that if $g$ is a left-invariant $\mathcal{J}$-Hermitian metric on $G_2$ such that $g$ is invariant under the right action of $T$ and for which $(g,\mathcal{J})$ is SKT, then $g$ must belong to this 3-parameter family of left-invariant SKT metrics. |
| title | A family of left-invariant SKT metrics on the exceptional Lie group $G_2$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2407.10037 |