A tractable definition of the homogeneous nearly K & auml;hler structure on (CP3 is given via the Hopf fibration, facilitating explicit computations and analysis. The description extends to all homogeneous metrics on (C P3, providing expressions for their Riemann curvature tensors and full isometry groups. Rigid immersions are presented for all extrinsically homogeneous Lagrangian submanifolds in the nearly K & auml;hler (C P3, and the non-existence of Lagrangians with constant sectional curvature is established. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we show how to construct a special class of ruled hypersurfaces in the nonflat complex space forms CPn and CHn. This is done by taking an arbitrary smooth curve in a totally geodesic (complex) one-dimensional submanifold and erecting an orthogonal ruling over each of its points. Concentrating on the n=2 case, we also examine the special situation in which the base curve has constant geodesic curvature. We show that, in this case, the construction yields precisely the real-analytic hypersurfaces of cohomogeneity one that satisfy certain transversality and nonvanishing conditions.
We investigate left-invariant statistical structures on Lie groups whose symmetric statistical tensor is generated by a single 1-form. Indeed, the symmetric part of the connection is given by μS=ω⊗Id+Id⊗ω, where Id is the identity operator. This condition produces a natural rank-one deformation of the Levi–Civita connection. We derive an algebraic compatibility condition between the Lie bracket and the defining 1-form that characterizes when this construction yields a statistical Lie group. Structural properties of the distinguished vector field U=ω♯ are obtained, including the fact that U is geodesic, and the form defining the structure is determined explicitly for Lie groups of Milnor type I. Explicit formulas are established for the cubic tensor, the Tchebychev vector field, and the statistical curvature tensor. Furthermore, we explicitly compute the statistical Laplacian and provide a geometric interpretation of its intrinsic thermodynamic drift across different dimensions of hyperbolic manifolds, illustrating how statistical drift interacts with background geometry. Finally, we formally define statistical vector fields and establish that these statistical Lie groups modeled by 1-forms exhibit extreme geometric rigidity. Specifically, the resulting almost-Abelian structures admit no non-trivial left-invariant Killing vector fields. We prove that this rigidity' phenomenon is entirely independent of the choice of the left-invariant Riemannian metric, arising purely from the underlying algebraic constraints. We also establish a Bochner-type identity for the statistical Laplacian and a fundamental commutation theorem showing that these continuous symmetries commute with the statistical Laplacian, thereby preserving its eigenspaces and ensuring the symmetry of the statistical heat flow.
We develop the theory of spinorial polyforms associated with bundles of irreducible Clifford modules of non-simple real type, obtaining a precise characterization of the square of an irreducible real spinor in signature (p-q) =8 1 as a polyform belonging to a semi-algebraic real set. We use this formalism to study general parallel spinors on Lorentzian three-manifolds, proving that in this dimension and signature, every parallel spinor is equivalent to an isotropic line preserved in a given direction by a metric connection with prescribed torsion. We apply this result to investigate Lorentzian three-manifolds equipped with a skew-torsion parallel spinor, namely a spinor parallel with respect to a metric connection with totally skew-symmetric torsion. We obtain several structural results about this class of Lorentzian three-manifolds, which are necessarily Kundt, and in the compact case, we obtain an explicit differential condition that guarantees geodesic completeness. We further elaborate on these results to study the supersymmetric solutions of three-dimensional NS-NS supergravity, which involve skew-torsion parallel spinors whose torsion is given by the curvature of a curving on an abelian bundle gerbe. In particular, we obtain a correspondence between NS-NS supersymmetric solutions and null coframes satisfying an explicit exterior differential system that we solve locally. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
