arXiv subjects

José Luis Carmona Jiménez

Publications and source records attributed to José Luis Carmona Jiménez.

7 recordsLinked to original sources

arXiv2026

Complex generalized spin-c Killing spinors in low dimensions

We classify all complex generalized spin-c Killing spinors with complex Killing function $iμ-α$ on complete Riemannian surfaces, presenting an explicit list of all possible cases. We show that, contrary to previous statements in the literature, there exist complex generalized spin-c Killing spinors on Riemannian surfaces for which $αμ\not\equiv0$. We illustrate the classification by providing several explicit local, compact, and complete examples, including a complete example for which $αμ>0$ everywhere. We then move to three dimensions, proving that a spin-c Riemannian three-manifold admits a complex generalized spin-c Killing spinor with complex Killing function $iμ-α$ if and only if it is a $μ$-exact trans-Sasakian manifold of type $(α,μ)$. In addition, we construct a geodesically incomplete Riemannian three-manifold equipped with a complex generalized spin-c Killing spinor for which neither $μ$ nor $α$ vanishes identically, and then prove that this is not possible if the characteristic vector field of the associated trans-Sasakian structure is complete. These results settle a conjecture by Große and Nakad (Results in Mathematics, 2015) stating that on two- and three-dimensional Riemannian manifolds the complex Killing function of any complex generalized spin-c Killing spinor must be either globally real or globally imaginary.

math.DG↗

arXiv2026

Kenmotsu manifolds and spin-c Killing spinors

Using the theory of complex spinorial forms, we prove that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with an imaginary Killing function $iμ$ if and only if it is an exact $μ$-Kenmotsu manifold, thereby obtaining an extension of a recent result by the first named author that, under the purity assumption, does not require simple connectivity or completeness. We then reinterpret $μ$-Kenmotsu manifolds in terms of metric connections with vectorial torsion and give a second proof of this characterization, combining the theory of complex spinorial forms with the theory of metric connections with torsion. Finally, we describe the global structure of exact $μ$-Kenmotsu manifolds by means of Morse-Bott theory.

math.DG↗

arXiv2026

On generalized imaginary $\mathrm{Spin}^c$-Killing spinors

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.

math.DG↗

arXiv2026

On Weyl structures reducible in the direction of the Lee form

A Weyl structure on a Riemannian manifold $(M,g)$ is a torsion-free linear connection $\nabla$ such that there is a $1$-form $θ$ (called the Lee form) satisfying $\nabla g = 2\, θ\otimes g$. We examine the case in which there exists a $\nabla$-parallel distribution of codimension $1$ on which the Lee form vanishes identically. We prove that if $(M,g)$ is complete with $θ$ closed, then the Weyl structure must be flat or exact. We apply this to prove the conjecture of Lotta (Eur. J. Math., 2023), namely, every homogeneous Kenmotsu manifold is isometric to the real hyperbolic space.

math.DG↗

arXiv2025

Canonical Reductive Decomposition of Extrinsic Homogeneous Submanifolds

Let $\overline{M}=\overline{G}/\overline{H}$ be a homogeneous Riemannian manifold. Given a Lie subgroup $G\subset \overline{G}$ and a reductive decomposition of the homogeneous structure of $\overline{M}$, we analyze a canonical reductive decomposition for the orbits of the action of $G$. These leaves of the $G$-action are extrinsic homogeneous submanifolds and the analysis of the reductive decomposition of them is related with their extrinsic properties. We connect the study with works in the literature and initiate the relationship with the Ambrose-Singer theorem and homogeneous structures of submanifolds.

math.DG↗

arXiv2024

The Ambrose-Singer Theorem for cohomogeneity one Riemannian manifolds

We characterize isometric actions whose principal orbits are hypersurfaces through the existence of a linear connection satisfying a set of covariant equations in the same spirit as the Ambrose-Singer Theorem for homogeneous space. These results are then used to describe isometric cohomogeneity one foliations in terms of such connections. Finally, we provide explicit examples of these objects in Euclidean spaces and real hyperbolic spaces.

math.DG↗

arXiv2021

The homogeneous geometries of complex hyperbolic space

We describe the holonomy algebras of all canonical connections and their action on complex hyperbolic spaces $\mathbb{C}\mathrm{H}(n)$ in all dimensions ($n\in\mathbb{N}$). This thorough investigation yields a formula for all Kahler homogeneous structures on complex hyperbolic spaces. Finally, we have related the belonging of the homogeneous structures to the different Tricerri and Vanhecke's (or Abbena and Garbiero's) orthogonal and irreducible $\mathrm{U}(n)$-submodules with concrete and determined expressions of the holonomy.

math.DG↗