The Carrollian superplane is constructed as a supermanifold generalization of the Carrollian plane.
problem Constructing the Carrollian superplane as a supermanifold.
method Intrinsic construction of the Carrollian superplane as a supermanifold generalization of the Carrollian plane, defining Carroll spinors, and showing it as a principal R1∣2-bundle. result Novel N=2 Carrollian supersymmetry transformations are generated. Hopf algebra structures on the extended q-superplane and its differential algebra are defined. An algebra of forms which are obtained from the generators of the extended q-superplane is introduced and its Hopf algebra structure is given
Introduces Carrollian Lie algebroids to handle singular Carrollian geometries.
problem Handling singular Carrollian geometries within standard Carrollian geometry.
method Introduces Carrollian Lie algebroids to study singular Carrollian geometries.
result Established the existence of compatible connections on Carrollian Lie algebroids.
A non-commutative differential calculus on the h-superplane is presented via a contraction of the q-superplane. An R-matrix which satisfies both ungraded and graded Yang-Baxter equations is obtained and a new deformation of the (1+1) dimensional classical phase space (the super-Heisenberg algebra) is introduced.
We introduce a noncommutative differential calculus on the two-parameter h-superplane via a contraction of the (p,q)-superplane. We manifestly show that the differential calculus is covariant under GLh1,h2(1∣1) transformations. We also give a two-parameter deformation of the (1+1)-dimensional phase space alge…
Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.
problem Developing Hodge theory on Carrollian bundles with degenerate metrics.
method Constructing Hodge star and Laplacian operators on Carrollian Rimes-bundles. result Hodge--de Rham Laplacian defined on event horizons of Schwarzschild black holes.
New framework for noncommutative Carrollian geometry using Lie-Rinehart pairs.
problem Developing a geometric framework for ultra-relativistic physics in noncommutative settings.
method Using ρ-Lie-Rinehart pairs to generalize Carrollian Lie algebroids to almost commutative geometry.
result Foundational principles of Carrollian geometry hold in almost commutative geometry.
In this work, differential geometry of the Z3-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.
Researchers compute differential invariants for Carrollian spacetimes.
problem Understanding the geometry and symmetries of Carrollian spacetimes.
method Derived from the geometry of the screen bundle, computed differential invariants using jet-spaces and Spencer cohomology.
result Specified how to generate the entire algebra of differential invariants for generic Carrollian structures, focusing on dimension 3.
Differential geometry of the quantum Lie superalgebra of the extended quantum superplane and its Z2-graded Hopf algebra structure is obtained. Its Z2-graded dual Hopf algebra is also given.
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
The paper explores geometric invariants of null hypersurfaces using Carrollian geometry.
problem Understanding the thermodynamics of black hole solutions.
method Examining various Carrollian geometries and their connections to null hypersurface embeddings.
result A connection with torsion is the most natural object to study Carrollian manifolds.
New approach to Carrollian geometry using Rimes-bundles.
problem Analyzing Carrollian manifolds with degenerate metrics.
method Principal Rimes-bundles with degenerate metrics and connections. result Canonical non-degenerate metric derived from principal connection.
The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.
problem Understanding projective and Carrollian geometries at infinity for Ricci flat Einstein manifolds.
method Developed a new type of Cartan geometry based on non-effective homogeneous models for projective geometry.
result Carrollian geometries are determined by the projective compactification data of Ricci flat Einstein manifolds.
New spaces at infinity identified for Minkowski spacetime.
problem Characterizing asymptotic infinities of Minkowski spacetime.
method Embedding and describing homogeneous spaces of the Poincaré group.
result Determined new structures on asymptotic infinities.
Gauging procedure constructs lagrangians for carrollian gravity.
problem Constructing lagrangians for carrollian gravity.
method Gauging procedure applied to Klein pairs corresponding to homogeneous spaces.
result Generalizes first-order lagrangians for four-dimensional maximally symmetric carrollian spaces.
The paper classifies intrinsic torsion in various spacetime structures.
problem Classifying intrinsic torsion in different spacetime structures.
method Review and classification of intrinsic torsion in galilean, Carrollian, Aristotelian, and Bargmannian spacetime structures.
result Found 16 classes for Aristotelian structures and 27 for Bargmannian structures.
Study of potential Carroll structures and special Carrollian manifolds for null hypersurfaces.
problem Understanding intrinsic geometry of null hypersurfaces.
method Initiate study of potential Carroll structures and explore their relationship to special Carrollian manifolds.
result Initiate the study of potential Carroll structures and their relationship to special Carrollian manifolds.
We define a new type of manifold and show it has properties like a pseudo-Riemannian manifold.
problem Defining a new type of manifold.
method Defining a Grassmann odd analogue of a Carrollian manifold and analyzing its properties.
result The reduced manifold is a pseudo-Riemannian manifold and compatible affine connections always exist with torsion.
Study p-brane Galilean and Carrollian geometries via intrinsic torsion.
problem Characterize p-brane Galilean and Carrollian geometries via intrinsic torsion. method Analyze intrinsic torsions as representations of G, interpret geometrically, and use physics-inspired methods. result Recover classification of p-brane Galilean geometries and relate to (D−p−2)-brane Carrollian geometries. The two-parametric quantum deformation of the algebra of coordinate functions on the supergroup GL(1∣1) via a contraction of GLp,q(1∣1) is presented. Related differential calculus on the quantum superplane is introduced.
Simply-connected homogeneous spacetimes for kinematical and aristotelian Lie algebras (with space isotropy) have recently been classified in all dimensions. In this paper, we continue the study of these "maximally symmetric" spacetimes by investigating their local geometry. For each such spacetime and relative to expon…
We classify simply-connected homogeneous (D+1)-dimensional spacetimes for kinematical and aristotelian Lie groups with D-dimensional space isotropy for all D≥0. Besides well-known spacetimes like Minkowski and (anti) de Sitter we find several new classes of geometries, some of which exist only for D=1,2. Th…