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.
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.
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.
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.
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.
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.
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.
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.
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. 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.
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.
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.
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. 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.
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 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.
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…
Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.
problem Characterize surfaces with constant ratio of principal curvatures in different geometries.
method Differential geometry, line geometry, Lie sphere geometry, ordinary differential equations, algebraic geometry.
result Characterized various types of surfaces like rotational, channel, ruled, helical, and translational.
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwe…
We define (p,q) hermitian geometry as the target space geometry of the two dimensional (p,q) supersymmetric sigma model. This includes generalised Kähler geometry for (2,2), generalised hyperkähler geometry for (4,2), strong Kähler with torsion geometry for (2,1) and strong hyperkähler with torsion geometry f…
Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.
problem Understanding non-lorentzian spacetimes.
method Classification and characterization of kinematical Lie algebras and their geometries.
result Characterization of Cartan geometries based on intrinsic torsion.
Simpler method derived for path geometries on surfaces, characterizing projective path geometries.
problem Characterizing projective path geometries on surfaces.
method Solving the equivalence problem of sub-Riemannian geometry of signature (1,1) on a contact 3-manifold.
result Characterization of projective path geometries in terms of their chains.
New definition of Born geometry connects to known geometries.
problem Defining and understanding Born geometries.
method Using Künneth structures and recursion operators.
result Born connection derived from Künneth connection for integrable geometries.
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
The paper extends group constructions to coset geometries, creating new ways to combine geometries.
problem Combining and gluing incidence geometries in a general framework.
method Extending classical group-theoretic constructions to coset geometries.
result Provides a general framework for combining or gluing incidence geometries.
Survey explores interactions between convex and complex geometry.
problem Understanding intersections between convex and complex geometry.
method Survey and review of existing literature.
result Demonstrates fascinating interactions between convex and complex geometry.
New symmetries found in Riemann-Cartan geometries.
problem Investigating symmetries in geometries with curvature and torsion.
method Mathematical tools to determine symmetries and subclasses of geometries.
result Determined all static and stationary spherically symmetric Riemann-Cartan geometries and subclasses with specific symmetries.
Lecture notes on geodesics in differential geometry.
problem Understanding geodesics in differential geometry.
method Expository lecture notes with exercises.
result Explains the geometry of geodesics.
Lecture notes on Finslerian geometry.
problem No specific problem stated; covers Finslerian geometry.
method Lecture notes.
result No specific key result mentioned.
Spin(7) geometry linked to multisymplectic geometry.
problem Understanding Spin(7) structures through multisymplectic geometry.
method Utilized Spin(7) identities to prove non-degeneracy of Cayley four-form in multisymplectic context.
result Spin(7) geometry is a special case of multisymplectic geometry.
The extension functors between categories of Cartan geometries can be used to define different categories of Cartan geometries with additional morphisms. The Cartan geometries modeled on skeletons can be used for the description of such categories of Cartan geometries and therefore we develop the theory of Cartan geome…
A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…
Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.
problem Understanding the geometry of SPD matrices for machine learning.
method Proposes a generalized Bures-Wasserstein geometry parameterized by a symmetric positive definite matrix.
result The GBW geometry outperforms the BW geometry in machine learning applications.
We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type (SO(2,3),P12), where P12 is a Borel parabolic subgroup in SO(2,3). We discuss the relations of this geometry of a car with the geometry of circles in the plane (a low dimensional Lie sph…
The study sets limits on the complexity of Klein geometries.
problem Understanding the complexity of Klein geometries.
method Simple upper and lower bounds for the order of Klein geometries.
result Established upper and lower bounds for the order of Klein geometries.
Surveying probabilistic real algebraic geometry.
problem Classical problems in real algebraic geometry.
method Probabilistic perspective on classical topics.
result Modern approach to Hilbert's Sixteenth Problem.
Develops Weyl structures for path geometries, simplifying their study.
problem Complexity in studying path geometries using traditional differential geometry methods.
method Defines distinguished connections and Schouten tensor, proving their dependence on line bundle sections.
result Shows a smaller subclass of Weyl structures for path geometries, with interesting connections to BGG sequences.
Ray-marching method visualizes 8 Thurston geometries in real-time.
problem Accurately rendering and visualizing Thurston geometries in real-time.
method Ray-marching algorithms with theoretical framework for non-Euclidean geometries.
result Accurate interactive real-time views of Thurston geometries achieved.
The paper extends Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
problem Extending Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
method Extending the Ruh-Vilms theorem to Weitzenböck geometry, considering the flat geometry with torsion.
result The Laplacian of the Gauss map for hypersurfaces in Weitzenböck geometry is related to the mean curvature vector field.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Introduces a new geometry based on difference angles, showing unique properties.
problem Defining angles independently of circles or rotations.
method Axiomatic system for difference angles, defining new geometric constructs.
result Explicit confirmation of the concurrency of the parabolic Miquel configuration.
We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.
KT-geometry is the geometry of a Hermitian connection whose torsion is a 3-form. HKT-geometry is the geometry of a hyper-Hermitian connection whose torsion is a 3-form. We identify non-trivial conditions for a reduction theory for these types of geometry.
We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 3 of 3) classifies those in which the linear isotropy representation is nontrivial but reducible. Most of the resulting geometries are products. Some interesting examples include a countably infinite family of inequiv…
Authors discuss complex and non-Archimedean geometry, proving a conjecture.
problem Proving a version of the Yau--Tian--Donaldson conjecture for Kähler metrics.
method Relation between complex, analytic, and non-Archimedean geometry.
result Sketch of proof for Yau--Tian--Donaldson conjecture.
The target space geometry of abelian vector multiplets in N=2 theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometrie…
Quantum field theory connects Riemannian geometry to quantum fluctuations.
problem Generating Riemannian structures from quantum fluctuations.
method QFT approach to Riemannian Geometry, focusing on Ricci curvature.
result Ricci curvature is crucial in generating Riemannian structures.
New geometry based on Siegel upper half-space with volume formula.
problem Developing a new 3D geometry based on Siegel upper half-space.
method Constructing a geometry fibered over Siegel upper half-space and providing a volume formula.
result Volume of Siegel-Seifert closed manifolds is the fiber circle length times base manifold's Euler characteristic.