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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.07813 |
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| _version_ | 1866911662771011584 |
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| author | Jiménez, José Luis Carmona |
| author_facet | Jiménez, José Luis Carmona |
| contents | A non-trivial spinor field $ψ$ is called a generalized imaginary $\mathrm{Spin}^c$-Killing spinor if $\nabla^{g,A} _X ψ= iμX \cdot ψ$ for all vector fields $X$, where $μ$ is a real function that is not identically zero and $\nabla^{g,A}$ is the $\mathrm{Spin}^c$ Levi-Civita connection with $\mathrm{U}(1)$-connection $A$. Associated with $ψ$ is a vector field $V$, the Dirac current, defined by $g(V,X) = i \langle X\cdot ψ, ψ\rangle$. We prove that if $V$ vanishes somewhere and $\operatorname{dim} M \geq 3$, the manifold is locally isometric to real hyperbolic space. When $V$ never vanishes and $\operatorname{dim} M \geq 3$, we obtain a global geometric description of all $\mathrm{Spin}^c$-Riemannian manifolds carrying such spinors, under the assumption that either the normalized Dirac current $ξ= \frac{V}{|V|}$ is complete or the leaves of $\mathcal{D} = \ker(ξ^\flat)$ are complete. Finally, we reinterpret the case of type~I generalized imaginary $\mathrm{Spin}^c$-Killing spinors in terms of parallel spinors for a suitable connection with vectorial torsion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07813 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On generalized imaginary $\mathrm{Spin}^c$-Killing spinors Jiménez, José Luis Carmona Differential Geometry A non-trivial spinor field $ψ$ is called a generalized imaginary $\mathrm{Spin}^c$-Killing spinor if $\nabla^{g,A} _X ψ= iμX \cdot ψ$ for all vector fields $X$, where $μ$ is a real function that is not identically zero and $\nabla^{g,A}$ is the $\mathrm{Spin}^c$ Levi-Civita connection with $\mathrm{U}(1)$-connection $A$. Associated with $ψ$ is a vector field $V$, the Dirac current, defined by $g(V,X) = i \langle X\cdot ψ, ψ\rangle$. We prove that if $V$ vanishes somewhere and $\operatorname{dim} M \geq 3$, the manifold is locally isometric to real hyperbolic space. When $V$ never vanishes and $\operatorname{dim} M \geq 3$, we obtain a global geometric description of all $\mathrm{Spin}^c$-Riemannian manifolds carrying such spinors, under the assumption that either the normalized Dirac current $ξ= \frac{V}{|V|}$ is complete or the leaves of $\mathcal{D} = \ker(ξ^\flat)$ are complete. Finally, we reinterpret the case of type~I generalized imaginary $\mathrm{Spin}^c$-Killing spinors in terms of parallel spinors for a suitable connection with vectorial torsion. |
| title | On generalized imaginary $\mathrm{Spin}^c$-Killing spinors |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.07813 |