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.
Simplifies Riemann geometry computations without indices.
problem Complex computations in Riemann geometry.
method Uses forms with Clifford algebra values and a specific group action.
result Simplified computations in Riemann geometry.
SU(2) flat connections link to 3D geometry with cosmological constant.
problem Mapping flat connections on Riemann surfaces to 3D twisted geometry.
method Relating flat connection quantities to geometrical quantities in discrete 3D space.
result Moduli space of SU(2) flat connections generalizes phase space of twisted geometry.
Study connects Riemann-Finsler geometry to Lorentz-violating scalar fields.
problem Exploring the connection between Riemann-Finsler geometries and Lorentz-violating scalar fields.
method Deriving quadratic actions and classical relativistic point-particle lagrangians in various spacetime dimensions.
result Support for open conjectures about Riemann-Finsler geometries in Lorentz-violating field theories.
Book reviews Riemann's work's impact on math, philosophy, physics.
problem None explicitly stated, focuses on Riemann's contributions.
method Review of Riemann's work and its impact.
result Riemann's work has broad impacts on math, philosophy, physics.
Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.
problem Addressing surfaces in Riemann-Cartan geometry with nontrivial torsion.
method Introducing a complex-valued 2-form associated with the torsion, which interacts with other geometric concepts.
result Complex-valued mean curvature quantity interacts with Hopf differential and Gauss map.
The study examines Fisher-Riemann geodesics for nonparametric probability densities.
problem Understanding nonparametric probability densities using Fisher-Riemann geometry.
method Obtaining Fisher-Riemann geodesics as a limit of parametric cases with increasing parameters.
result The weak limit approach for nonparametric probability densities.
Super Riemann surfaces extend Riemann surfaces with an additional field, the gravitino.
problem Extending the study of Riemann surfaces to include supergeometry.
method Presenting an extension of the harmonic action functional to super Riemann surfaces.
result Super Riemann surfaces can be studied using an extended harmonic action functional.
Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a …
In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi su…
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.
problem Cohomology of the Regge complex in three dimensions.
method Constructing a discrete version of linearized Riemann-Cartan geometry on any triangulation.
result The cohomology of the Regge complex is isomorphic to the infinitesimal-rigid-body-motion-valued de~Rham cohomology.
Bipartite Riemann-Finsler geometries with complementary Finsler structures are constructed. Calculable examples are presented based on a bilinear-form coefficient for explicit Lorentz violation.
Study Kähler geometry on Hurwitz spaces of Riemann surfaces.
problem Curvature of Kähler metric on Hurwitz spaces.
method Generalized Weil-Petersson metric and Horikawa's deformation theory.
result Investigate curvature of Kähler metric on Hurwitz spaces.
We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.
Lectures introduce differential geometry for holomorphic vector bundles on Riemann surfaces.
problem Analyzing holomorphic vector bundles on Riemann surfaces.
method Differential-geometric techniques.
result Introduction of geometric methods for studying these bundles.
We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds.
Characterizes conformal classes of tori using differential geometry.
problem Classifying conformal classes of tori in complex dimension 1.
method Basic differential geometry methods, contrasting with Hopf tori.
result Complete characterization of conformal classes of product and standard flat tori.
The study describes how maps between Riemann surfaces evolve under mean curvature flow.
problem Evolution of area decreasing maps between Riemann surfaces under mean curvature flow.
method Mean curvature flow applied to area decreasing maps between Riemann surfaces with bounded geometry.
result Complete description of the evolution of area decreasing maps.
Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.
problem Defining and studying invariants of elliptic curves in locally conformally symplectic manifolds.
method Using J-holomorphic curves and Gromov-Witten theory to define and study invariants. result Found new phenomena in Riemann-Finsler geometry and an analogue of the Weinstein conjecture.
Riemann's math ideas often come from physics, blending philosophy and science.
problem Separating Riemann's mathematical ideas from their physical origins.
method Overview of Riemann's work, emphasizing physical motivation and philosophical context.
result Riemann's mathematical results are deeply intertwined with physical reasoning and philosophical ideas.
The paper studies new curvature properties in Finsler geometry.
problem Properties of projectively equivalent Finsler metrics and their curvature structures.
method Introducing new characterizations of quadratic curvature properties in Finsler manifolds.
result Novel insights into curvature behavior under generalized projective sprays.
The work extends the A. Connes' noncommutative geometry to spaces with generic local anisotropy. We apply the E. Cartan's anholonomic frame approach to geometrical models and physical theories and develop the nonlinear connection formalism for projective module spaces. Examples of noncommutative generation of anholonom…
New method to parametrize infinite Riemann surfaces with bounded triangulations.
problem Parametrizing infinite Riemann surfaces with bounded triangulations.
method Introducing bounded ideal triangulations and proving real-analyticity of the parametrization.
result Real-analytic parametrization of Teichmüller spaces for infinite surfaces with bounded triangulations.
Overview of 19th century innovations leading to complex geometry.
problem Understanding the origins of complex geometry in the 19th century.
method Detailed examination of key papers and theories from the 19th century.
result Key innovations in the 19th century led to complex geometry in the 20th century.
We study the spectral geometry of the Riemann curvature tensor for Pseudo-Riemannian manifolds and provide some examples illustrating the phenomena which can arise in the higher signature setting. Dedication: This paper is dedicated to the memory of our colleague Prof.G. Tsagas who studied the spectral geometry of Lapl…
Survey of early complex analysis and topology, highlighting Euler's influence on Riemann's work.
problem Understanding the historical development of complex analysis and topology.
method Historical review of key mathematical concepts and Euler's contributions.
result Euler's foundational work influenced Riemann's seminal discoveries in complex analysis and topology.
Paper connects volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.
problem Relating volumes of moduli spaces of super Riemann surfaces to integrals over stable Riemann surfaces.
method Relates volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces Mg,n. result Proves recursion between volumes of moduli spaces of super hyperbolic surfaces using algebraic geometry.
Physical reasons suggested in \cite{Ha-Ha} for the \emph{Quantum Gravity Problem} lead us to study \emph{type-changing metrics} on a manifold. The most interesting cases are \emph{Transverse Riemann-Lorentz Manifolds}. Here we study the conformal geometry of such manifolds.
RSVGD improves SVGD for Bayesian inference on Riemannian manifolds.
problem Bayesian inference on Riemannian manifolds.
method Develops RSVGD, a generalization of SVGD to Riemann manifolds.
result Advantages over SVGD in exploring distribution geometry and particle-efficiency.
Abstract: Generalizes Riemann's bilinear relations for surfaces of genus ≥ 2.
problem No new significant results, but highlights a known observation.
method Generalization of Riemann's bilinear relations.
result No significant new results obtained.
The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.
problem Discrete conformal geometry of polyhedral surfaces.
method Establishing rigidity for hexagonal triangulations and estimating quasiconformal constants.
result Discrete conformal maps converge to Riemann mappings for Jordan domains.
We describe the (complex) quaternionic geometry encoded by the embeddings of the Riemann sphere, with nonnegative normal bundles.
We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry.
New tensors reveal full curvature structure from Riemann tensor.
problem Limited information from Ricci contraction of Riemann tensor.
method Contracting double dual of Riemann tensor to reveal full curvature.
result New tensors provide canonical parents of Einstein tensor.
A geometric approach to differential game theory is illustrated. The parallel pursuit is considered as a two-player zero-sum differential game. The optimal strategies of each player is designed based on Riemann-Finsler geometry. Our approach incorporates a closed loop optimal control and the presentation is familiar wi…
This article provides a brief discussion of the functional of super Riemann surfaces from the point of view of classical (i.e. not "super-) differential geometry. The discussion is based on symmetry considerations and aims to clarify the "borderline" between classical and super differential geometry with respect to the…
We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metr…
The study finds infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.
problem Existence and properties of semi-arithmetic Riemann surfaces.
method Combining number theory and hyperbolic geometry to prove existence and properties of semi-arithmetic Riemann surfaces.
result Existence of infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.
In this paper, we study the asymptotic geometry of Teichmuller space of Riemann surfaces and give bounds on the Weil-Petersson sectional curvature of Teichmuller space, in terms of the length of the shortest geodesic on the surface. This will also imply that the sectional curvature is not pinched from above or below by…
The paper studies geodesics on an icosahedron's Riemann surface.
problem Geodesics on a icosahedron's Riemann surface.
method Constructs a Riemann surface and analyzes geodesics on it.
result Discovers properties of geodesics under a flat metric.
The study explores autonomous systems and their connections to contact geometry and Frobenius manifolds.
problem Understanding the connections between autonomous systems and geometric structures.
method Investigation of the Darboux-Halphen-Ramanujan system, contact geometry, and Frobenius manifolds.
result Highlighting the role of contact geometry in autonomous systems.
Harmonic maps connect physics, geometry, and analysis.
problem Understanding singularities in geometric analysis.
method Analysis of harmonic maps and their supersymmetric extensions.
result Formation of bubbles in geometric analysis.
Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.
problem Understanding projective flat holomorphic vector bundles over Riemann surfaces.
method Assigning Wronskian line bundles to vector bundles and interpreting Abel's identity.
result Abel's identity is the first Chern class of the Wronskian line bundle.
Study of surfaces with marked horizontal separatrices on Riemann surfaces.
problem Counting and classifying surfaces with specific geometric structures.
method Computing connected components and using topological invariants.
result At most two components for surfaces with genus greater than 0, except in hyperelliptic cases.
In this paper, it is elaborated the theory the Ricci flows for manifolds enabled with nonintegrable (nonholonomic) distributions defining nonlinear connection structures. Such manifolds provide a unified geometric arena for nonholonomic Riemannian spaces, Lagrange mechanics, Finsler geometry, and various models of grav…
The aim of this paper is to construct a natural Riemann-Lagrange differential geometry on 1-jet spaces, in the sense of nonlinear connections, generalized Cartan connections, d-torsions, d-curvatures, jet electromagnetic fields and jet electromagnetic Yang-Mills energies, starting from some given nonlinear evolution OD…
Formula calculates Riemann-Roch number for singular symplectic quotients.
problem Computing Riemann-Roch number for singular symplectic quotients.
method Complete singular stationary phase expansion of Witten integral.
result New explicit local invariant of singularities in symplectic quotients.
The paper generalizes Cartan Geometry using Polacek and Siegel's approach.
problem Formulating sigma model dynamics in a covariant way.
method Using Polacek and Siegel's generalised curvature and torsion approach within the generalised metric formalism.
result Almost all higher generalised tensors correspond to covariant derivatives of the generalised Riemann tensor.