Talks about new methods in differential geometry.
problem None explicitly stated in the abstract.
method Stresses the neighbor relation as a basic notion.
result Not explicitly stated in the abstract.
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48 results for “differential-geometry”
Explains curves and surfaces in differential geometry.
problem Understanding smooth curves and surfaces in differential geometry.
method Problem-centered, elementary, visual approach focusing on essential techniques.
result Provides a solid foundation for further study in differential geometry.
These notes introduce key techniques in differential geometry for curves and surfaces.
problem Understanding the basics of differential geometry for curve and surface analysis.
method Problem-centered, elementary, visual approach to teaching essential techniques.
result Provides a solid foundation for further study in differential geometry.
In this paper is proposed a kind of model theory for our axiomatic differential geometry. It is claimed that smooth manifolds, which have occupied the center stage in differential geometry, should be replaced by functors on the category of Weil algebras. Our model theory is geometrically natural and conceptually motiva…
Abstract: Explains translating derived algebraic geometry results to derived differential geometry.
problem Existence and classification of deformation quantizations on NQ-manifolds.
method Translation of results from derived algebraic geometry to derived differential geometry.
result Existence and classification of various deformation quantizations.
Lecture notes introduce differential geometry using sheaves and differential operators.
problem Exploring differential geometry concepts.
method Using sheaves, differential operators, and horizontal subbundles.
result Presented an approach to fundamental differential geometry structures.
Quaternionic differential geometry expands geometric concepts using quaternions.
problem Generalizing geometric concepts to quaternionic constraints.
method Generalizing curves and surfaces, curvature, torsion, differential forms, and directional derivatives to quaternionic constraints.
result Quaternionic formalism provides a suitable language for differential geometry.
The paper explores quaternionic curves using differential geometry.
problem Understanding quaternionic curves.
method Differential geometry applied to quaternionic curves.
result Simpler formulations of quaternionic curves.
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra)…
In our previous paper entitled "Axiomatic differential geometry -towards model categories of differential geometry-, we have given a category-theoretic framework of differential geometry. As the first part of our series of papers concerned with differential-geometric developments within the above axiomatic scheme, this…
We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.
Analytic proof solves differential geometry problem.
problem Solving the system of equations , in connected domains.
method Analytic approach to solve differential equations.
result Validated Segre's Theorem in differential geometry.
Formalizes synthetic differential geometry in Lean.
problem Formalizing synthetic differential geometry in a proof assistant.
method Formalization of synthetic differential geometry with Lean and mathlib.
result Proves a Taylor theorem for functions of several variables.
Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avant-garde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framewor…
We give a brief review of a research made in the field of differential geometry in Estonia in the period from the beginning of the 19th century to the present time. The biographic data of mathematicians who made a valuable contribution to the development of differential geometry in Estonia in mentioned period are prese…
The paper uses complex-valued functions to simplify plane differential geometry and kinematics.
problem Simplifying complex problems in plane differential geometry and kinematics.
method Consistent use of complex-valued functions of a real variable.
result Derives results in a particularly simple, uniform, and transparent way.
Symmetry-breaking in three differential geometry conjectures.
problem Exploring the role of symmetry in three differential geometry conjectures.
method Examining the Carathéodory, Willmore, and Lawson Conjectures through the lens of symmetry in 3D space-forms.
result Symmetry is broken, and more general ambient metrics are considered, leading to the failure of the conjectures.
The relation between differential geometry of surfaces and some Heisenberg ferromagnet models is considered.
Algebraic geometry replaces manifolds in differential geometry.
problem Eliminate the need for manifolds in differential geometry.
method Introduce algebraifolds and use commutative algebras with finitely generated projective module of derivations.
result General relativity can be formulated using algebraifolds.
A connection between differential geometry and soliton equations is discussed
Paper connects differential geometry with geometric calculus.
problem None explicitly stated in the abstract.
method Introduces a more general Laplacian for multivector-valued functions on manifolds.
result Formulated a higher codimensional analog of Jacobi's field equation.
This book is a textbook for the basic course of differential geometry. It is recommended as an introductory material for this subject.
Global homotopies upgrade classical map in differential geometry.
problem Upgrade classical Hochschild-Kostant-Rosenberg map to a deformation retract.
method Combining symbol calculus and coalgebraic van Est theorem.
result Develop deformation retracts in various settings.
Why Diffeology?math.DG
Diffeology extends differential geometry to complex spaces.
problem Handling singular and infinite-dimensional settings in differential geometry.
method Introduces diffeology as a new framework.
result Diffeology provides a natural and effective framework for complex spaces.
In this paper we give an axiomatization of differential geometry comparable to model categories for homotopy theory. Weil functors play a predominant role.
Submanifold theory is a very active vast research field which plays an important role in the development of modern differential geometry. This branch of differential geometry is still so far from being exhausted; only a small portion of an exceedingly fruitful field has been cultivated, much more remains to be discover…
Geometry of geodesicsmath.DG
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.
PINNs solve differential geometry problems in complex shapes.
problem Solving differential geometry problems in complex shapes.
method Training neural networks with loss functions inspired by differential conditions.
result PINNs are effective for differential geometry problems.
Report on formalizing differential geometry in Lean.
problem Formalizing differential geometry in a proof assistant.
method Lean's type theory approach to formalization.
result Surprising differences between formal and informal proofs.
Extends differential geometry concepts to manifolds with super tangent bundles.
problem No specific problem stated; extending differential geometry to super tangent bundles.
method Introduces super tangent bundle and extends differential geometry concepts.
result Basic notions of differential geometry extended to manifolds with super tangent bundles.
In this paper, as the second in our series of papers on differential geometry of microlinear Frolicher spaces, we study differenital forms. The principal result is that the exterior differentiation is uniquely determined geometrically, just as grad (ient), div (ergence) and rot (ation) are uniquely determined geometric…
This book teaches differential geometry of curves and surfaces in 3D Euclidean space.
problem Understanding differential geometry concepts for mathematics and sciences students.
method Covering definitions, ideas, and concepts with examples and illustrations.
result Successful understanding and application of differential geometry.
In this work, differential geometry of the Z-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.
Projective geometry aids in analyzing fields near compact manifolds.
problem Analyzing fields near compact manifolds.
method Developed a projective exterior differential tractor calculus.
result Analogous calculus for projectively compact manifolds.
Segre embedding was introduced by C. Segre (1863--1924) in his famous 1891 article \cite{segre}. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathematical physics, and coding theory. In this article, we survey main results on Segre embedding in differential geo…
Using standard analysis only, we present an extension of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesim…
Quantum complexity lowerbound proved using differential geometry.
problem Proving lower bounds on quantum complexity.
method Applied the Bishop-Gromov bound to Nielsen's complexity geometry.
result Lower bounds on quantum complexity are exponentially large.
Abstract: Review of Index theorem and its applications.
problem Index theorem applications in differential geometry.
method Review of Atiyah-Singer Index theorem.
result Basic knowledge required for understanding.
Comparison theorems in centro-affine differential geometry
problem rigidity phenomena of comparison theorems
method study of centro-affine differential geometry
result examples of rigidity phenomena
Simplified Beltrami's theorem using geometry.
problem Beltrami's theorem
method Parabolic differential geometry
result Simplified approach to Beltrami's theorem
The paper extends Bäcklund theorem in affine differential geometry of surfaces.
problem Understanding local symmetries and proportional affine fundamental forms in surfaces.
method Formulating and proving complementary affine Bäcklund theorem.
result Conditions for local symmetry and proportional affine fundamental forms.
Linearized Einstein equations simplified via Calabi operator.
problem Linearizing Einstein equations for cosmological applications.
method Using the Calabi operator from projective differential geometry.
result Linearized Einstein equations simplified.
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.
Differential geometry of the quantum Lie superalgebra of the extended quantum superplane and its Z-graded Hopf algebra structure is obtained. Its Z-graded dual Hopf algebra is also given.
Illustrates Ricci flow for a general audience.
problem Explains Ricci flow to non-experts.
method Illustrative introduction without prerequisites.
result Provides a working definition and intuition for Ricci flow.
New differential geometry perspective on orthogonal RNNs.
problem Mitigating exploding and vanishing gradients in RNNs.
method Using tools from differential geometry, parameterizing vector fields via directional derivatives of scalar functions.
result Our approach achieves comparable or better results on benchmark tasks.
Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.
A pictorial introduction to differential geometry, leading to Maxwell's equations as three picturesmath.DG
This article simplifies differential geometry concepts for physics students.
problem Understanding differential geometry for physics students.
method Presenting concepts through pictures, avoiding equations.
result Maxwell's equations presented as three pictures.