Differential calculus for generalized geometry and geometric Lax flows
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914966721789952 |
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| author | Hu, Shengda |
| author_facet | Hu, Shengda |
| contents | Employing a class of generalized connections, we describe certain differential complices $\left(\tilde Ω^*_{\mathbb{T}}(M), \tilde{\mathbb{d}}^{\mathbb{T}}\right)$ constructed from $\wedge^* \mathbb{T} M$ and study some of their basic properties, where $\mathbb{T} M = T M \oplus T^*M$ is the generalized tangent bundle on $M$. A number of classical geometric notions are extended to $\mathbb{T} M$, such as the curvature tensor for a generalized connection. In particular, we describe an analogue to the Levi-Civita connection when $\mathbb{T} M$ is endowed with a generalized metric and a structure of exact Courant algebroid. We further describe in generalized geometry the analogues to the Chern-Weil homomorphism, a Weitzenböck identity, the Ricci flow and Ricci soliton, the Hermitian-Einstein equation and the degree of a holomorphic vector bundle. Furthermore, the Ricci flows are put into the context of geometric Lax flows, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_04566 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Differential calculus for generalized geometry and geometric Lax flows Hu, Shengda Differential Geometry Employing a class of generalized connections, we describe certain differential complices $\left(\tilde Ω^*_{\mathbb{T}}(M), \tilde{\mathbb{d}}^{\mathbb{T}}\right)$ constructed from $\wedge^* \mathbb{T} M$ and study some of their basic properties, where $\mathbb{T} M = T M \oplus T^*M$ is the generalized tangent bundle on $M$. A number of classical geometric notions are extended to $\mathbb{T} M$, such as the curvature tensor for a generalized connection. In particular, we describe an analogue to the Levi-Civita connection when $\mathbb{T} M$ is endowed with a generalized metric and a structure of exact Courant algebroid. We further describe in generalized geometry the analogues to the Chern-Weil homomorphism, a Weitzenböck identity, the Ricci flow and Ricci soliton, the Hermitian-Einstein equation and the degree of a holomorphic vector bundle. Furthermore, the Ricci flows are put into the context of geometric Lax flows, which may be of independent interest. |
| title | Differential calculus for generalized geometry and geometric Lax flows |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2206.04566 |