Study on geodesic completeness for Type A surfaces.
problem Determining geodesic completeness for Type A surfaces.
method Analyzing constant Christoffel symbols to determine geodesic completeness.
result A method to determine if a set of constant Christoffel symbols can model a complete surface.
In this note we exhibit a constructive demonstration of the uniqueness of the Christoffel symbol.
Directly proves Brioschi formula for Gaussian curvature.
problem Express Gaussian curvature in terms of local coordinates.
method Elementary proof without Christoffel symbols.
result Directly derived Brioschi formula for Gaussian curvature.
New invariants found for a specific type of mapping in generalized Riemannian space.
problem Understanding transformations of Christoffel symbols in generalized Riemannian space.
method Analysis of third type almost geodesic mappings.
result Obtained new invariants that are projective parameters and tensors.
Derives a formula for the k-th covariant derivative of tensor fields.
problem Finding a formula for the k-th covariant derivative of tensor fields.
method Introducing symbols P and Q depending on Christoffel symbols, deriving a formula (3.1).
result Derives a formula for the k-th covariant derivative of tensor fields.
Study geodesics on Siegel-Jacobi ball with balanced metric.
problem Understanding geodesics on the Siegel-Jacobi ball.
method Determined Christoffel symbols, studied geodesic equations, calculated covariant derivatives.
result Explicit formulas for geodesics on Siegel-Jacobi ball.
Derives Levi-Civita connection formulas for specific geometries.
problem Determining geometric invariants of Lorentzian manifolds.
method Explicitly derives Christoffel symbols in terms of adapted frame fields.
result Formulas for geometric invariants of Lorentzian manifolds.
We generalize the notion of a Lie algebroid over infinite jet bundle by replacing the variational anchor with an N-tuple of differential operators whose images in the Lie algebra of evolutionary vector fields of the jet space are subject to collective commutation closure. The linear space of such operators becomes an a…
The article approximates solutions to the Beltrami equation using similarity surfaces.
problem Approximating solutions to the Beltrami equation.
method Constructing similarity surfaces from polygons and analyzing their conformal uniformization.
result Holomorphic dependence of Christoffel symbols on polygons and convergence to a specific affine connection.
In this paper, we study the inverse surfaces in 3-dimensional Euclidean space E3. We obtain some results relating Christoffel symbols, the normal curvatures, the shape operators and the third fundamental forms of the inverse surfaces
Generalized Lagrange-Weyl structures and compatible connections are introduced as a natural generalization of similar notions from Riemannian geometry. Exactly as in Riemannian case, the compatible connection is unique if certain symmetry conditions with respect to vertical and horizontal Christoffel symbols are impose…
Unified flow solves Lp Christoffel-Minkowski problem for p>1.
problem Solving the Lp Christoffel-Minkowski problem for p>1. method Anisotropic expanding flow of smooth hypersurfaces with speed ψσk(λ)α. result The flow converges to a solution of the Lp Christoffel-Minkowski problem. Geometrically represents path integral reduction Jacobian for interacting systems.
problem Quantizing a model mechanical system with dependent coordinates.
method Geometric representation using scalar curvature and Christoffel symbols in a nonholonomic basis.
result Found a geometric representation for the path integral reduction Jacobian.
We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour congruences is given. Surfaces of constant mean curvature are special among all iso…
Geometric calculus on probability simplex using Wasserstein metric.
problem Calculus on the probability simplex with Wasserstein metric.
method Embedding probability simplex in positive measure space with nonlinear metric tensor, deriving Christoffel symbols, connections, curvature tensors, and operators.
result Established geometric computations on probability manifold and density space, connecting Fisher-Rao and optimal transport metrics.
Proves equivalence of two types of Ricci curvature bounds.
problem Equivalence of distributional and synthetic Ricci curvature bounds.
method Analyzes weighted Riemannian manifolds with specific smoothness conditions.
result Proves equivalence of Ricci curvature bounds under given conditions.
We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a concentric quadric.
In the present article the geometry of semi-Riemannian manifolds with nonholonomic constraints is studied. These manifolds can be considered as analogues to the sub-Riemannian manifolds, where the positively definite metric is substituted by a nondegenerate metric. To study properties of the exponential map the Christo…
Paper studies inverse curvature flows and solves related geometric problems.
problem Inverse curvature flows and related geometric problems.
method Analyzes a class of expanding flows with specific speeds and proves existence and convergence.
result Proves the existence and convergence of flows under certain conditions, leading to new solutions to geometric problems.
In this paper, we define the inverse surface of a tangent developable surface with respect to the sphere S_{c}(r) with the center c∈E3 and the radius r in 3-dimensional Euclidean space E3. We obtain the curvatures, the Christoffel symbols and the shape operator of this inverse surface by …
We present the non-trivial example how to generate non-Euclidean geometries from associative unital algebras. We consider bundles of the sphere of the degenerate non-Eucleadian space and its two models. The first (conformal) model is obtained by the mapping S onto a plane pass through the origin. It is analogous to the…
Introduces canonical connections for sub-Riemannian manifolds with constant symbol.
problem Equivalence problem in sub-Riemannian geometry.
method Introduces canonical grading and compatible affine connection.
result Completely computed structures for contact manifolds of constant symbol.
Anisotropic expanding flow of convex hypersurfaces converges to a soliton under certain conditions.
problem Anisotropic expanding curvature flows of convex hypersurfaces in Euclidean space.
method Proving the existence and convergence of a unique smooth and uniformly convex solution to the flow under specific conditions.
result The flow converges to a soliton which solves an elliptic equation when parameters are within a suitable range.
New method calculates geodesic distances in Gaussian random field manifolds.
problem Quantifying similarity between random fields in different regimes.
method Numerical method using geodesic distances in Gaussian random field manifolds.
result Estimation of geodesic distances for various initial conditions.
Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from the Christoffel symbols. The procedure is extended for determining if a second …
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
The paper studies the geometry of probability measures on the unit circle.
problem Understanding the Riemannian geometry of probability measures on the unit circle.
method Developed an intrinsic framework using the Peter-Weyl Theorem to study the differential geometry of Wasserstein spaces of compact Lie groups.
result Explicitly demonstrated that the Wasserstein space of the unit circle is flat with vanishing curvature.
A new approach to Riemannian geometry using embedded and submersion structures.
problem Studying Riemannian geometry on manifolds embedded in Euclidean spaces.
method Identifying tangent bundles with subbundles of trivial bundles, extending metrics, and defining submersed ambient structures.
result Simplified formulas for Christoffel symbols and Riemannian curvature in embedded and submersion structures.
New flow method solves Christoffel-Minkowski problem.
problem Solving Christoffel-Minkowski problem.
method Entropy preserving curvature flow with global term.
result Entropy preserving flow solves Christoffel-Minkowski problem.
Solves capillary Lp-Christoffel-Minkowski problem in half-space.
problem Capillary surfaces in half-space geometry.
method Non-collapsing estimate for height and capillary support function.
result Extends Christoffel-Minkowski existence result.
Classifies 3-braids from choreographic motions on Lissajous curves, linking them to mapping classes and geodesics.
problem Classifying 3-braids from choreographic motions on Lissajous curves.
method Parametrization in terms of levels and slopes, using dilatation and geodesic cutting sequences.
result Dilatation of pseudo-Anosov mapping classes increases with level or slope.
A proof that hyperbolic plane cannot be immersed in Euclidean 3-space.
problem Proving the impossibility of isometrically immersing the hyperbolic plane in Euclidean 3-space.
method Applying ideas from undergraduate mathematics, including moving frames and connection forms, to simplify the proof.
result A key transition from principal directions to asymptotic directions simplifies the proof and yields a coordinate system.
Paper solves Christoffel-Minkowski problem in hyperbolic space.
problem Prescribing k-th horospherical p-surface area measure of h-convex domains in hyperbolic space. method Considered a fully nonlinear equation and used the full rank theorem with a viscosity approach.
result Existence of uniformly h-convex solution under appropriate assumptions. This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.
problem Prescribing curvatures for convex hypersurfaces in hyperbolic space.
method Proving a full rank theorem to establish the existence of solutions.
result Existence of solutions to the Christoffel problem and its equivalent Nirenberg-Kazdan-Warner problem on spheres.
Paper solves Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
problem Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
method Proved existence of solutions using a new full rank theorem.
result Existence of smooth, origin-symmetric, strictly horospherically convex solutions.
Study of hypersurfaces in curved spaces with specific curvature properties.
problem Characterizing hypersurfaces in spaces of constant curvature with particular curvature properties.
method Investigates hypersurfaces isometrically immersed in semi-Riemannian spaces of constant curvature, focusing on the curvature tensor and its properties.
result Hypersurfaces in the specified spaces satisfy a Roter type equation, linking their curvature tensor to specific tensor products.
Investigates the connection between quadrics and Christoffel duals, and zero mean curvature surfaces.
problem Understanding the relationship between quadrics and Christoffel duals, and zero mean curvature surfaces.
method Introducing para-holomorphic elliptic functions to study timelike minimal surfaces and their Christoffel duals of 1-sheeted hyperboloids.
result Curves of type change for real isothermic surfaces of mixed causal type are aligned with the real curvature line net.
A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvat…
Kernelized inverse Christoffel function improves outlier detection.
problem Outlier detection in large datasets with many features.
method Kernelized variant of the inverse Christoffel function.
result Best performance on 15 data sets compared to current methods.
Christoffel function characterizes the corruption a bounded-degree certificate cannot remove in robust halfspace learning.
problem Robust halfspace learning under malicious noise
method Sum-of-Squares degree of outlier-removal certificate
result Christoffel function bounds the corruption a bounded-degree certificate cannot remove
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the Lp-Christoffel-Minkowski problem. method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for c=1 under certain initial data. Study anisotropic flows without global terms and solve dual Orlicz Christoffel-Minkowski problems.
problem Anisotropic flows without global forcing terms and dual Orlicz Christoffel-Minkowski problems.
method Existence results for dual Orlicz Christoffel-Minkowski type problems via stationary solutions of anisotropic flows.
result Existence results for a class of dual Orlicz Christoffel-Minkowski type problems.
A geometric flow based in the Riemann-Christoffel curvature tensor that in two dimensions has some common features with the usual Ricci flow is presented. For n dimensional spaces this new flow takes into account all the components of the intrinsic curvature. For four dimensional Lorentzian manifolds it is found that…
Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.
problem Finding capillary convex bodies with prescribed k-th capillary area measure. method Solving a Hessian-type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution under natural conditions.
Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in Hn+1, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with t…
We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discre…
Researchers solve a specific case of the Lp Christoffel-Minkowski problem for 1<p<k+1.
problem Solving the Lp Christoffel-Minkowski problem for 1<p<k+1. method Establishing the existence of convex bodies with prescribed k-th even p-area measure under certain conditions. result Existence of convex bodies with prescribed k-th even p-area measure on Sn under appropriate assumptions. Solves Christoffel problem for disk area measures on spheres.
problem Conditions for a measure to be a disk area measure of convex bodies.
method Integral representation and differential equation reformulation.
result Reconstructs support function from disk area measure.