In this note we exhibit a constructive demonstration of the uniqueness of the Christoffel symbol.
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48 results for “Christoffel symbols”
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.
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.
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.
Type A surfaces are the locally homogeneous affine surfaces which can be locally described by constant Christoffel symbols. We address the issue of the geodesic completeness of these surfaces: we show that some models for Type A surfaces are geodesically complete, that some others admit an incomplete geodesic but model…
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 . 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…
We studied rules of transformations of Christoffel symbols under third type almost geodesic mappings in this paper. From this research, we obtained some new invariants of these mappings. These invariants are analogies of Thomas projective parameter and Weyl projective tensor.
We determine the Christoffel's symbols for the Siegel-Jacobi ball endowed with the balanced metric. We study the equations of geodesics on the Siegel-Jacobi ball. We calculate the covariant derivative of one-forms in the variables in which is expressed the balanced metric on the Siegel-Jacobi ball.
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.
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.
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…
Involutive distributions of operator-valued evolutionary vector fields and their affine geometry. IImath-ph
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…
In this paper, we define the inverse surface of a tangent developable surface with respect to the sphere S_{c}(r) with the center and the radius r in 3-dimensional Euclidean space . 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…
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.
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 …
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.
Christoffel-Minkowski flowsmath.DG
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.
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.
Solves capillary -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.
Lissajous 3-braidsmath.GT
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.
Paper solves Christoffel-Minkowski problem in hyperbolic space.
problem Prescribing -th horospherical -surface area measure of -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 -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.
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.
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the -Christoffel-Minkowski problem.
method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for 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 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 -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.
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.
Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.
problem Christoffel-Minkowski problem and Hessian equations under rotational symmetries.
method Constructing explicit convex solutions to mixed Monge-Ampère equations on \(\mathbb{R}^n\) under radial symmetry.
result Explicit representation formula for the support function of the resulting convex body.
Unified flow solves Christoffel-Minkowski problem for .
problem Solving the Christoffel-Minkowski problem for .
method Anisotropic expanding flow of smooth hypersurfaces with speed .
result The flow converges to a solution of the Christoffel-Minkowski problem.
Weyl and Cartan proposed different but related ways to handle infinitesimal geometry in the early 1920s.
problem How to apply transformation groups in differential geometry.
method Both used connections and parallel transfer, with Cartan aiming for a more general framework.
result They reached an agreement on handling Cartan's infinitesimal geometric structures by the 1930s.
Develops connections and curvature concepts for non-smooth spaces.
problem Exploring connections and curvature in spaces with singularities.
method Defines connections and curvature for Lie-Rinehart algebras without smoothness assumptions.
result Removal of poles in Christoffel symbols for quotient singularities.
Log-concave coefficient sequences for two-bridge knots proved.
problem Proving log-concavity of Alexander polynomial coefficient sequences for alternating knots.
method Introducing a polynomial associated to Christoffel words and proving its log-concavity.
result Strong Fox conjecture for two-bridge knots proved.
Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.
Paper solves a geometric problem involving mixtures of area and curvature measures.
problem Investigates a geometric problem involving mixtures of area and curvature measures.
method Establishes a gradient estimate to prove the existence of a solution.
result Proves the existence of an even, smooth, strictly convex solution for .
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.
It is shown that for small, spherically symmetric perturbations of asymptotically flat two-ended Reissner-Nordström data for the Einstein-Maxwell-real scalar field system, the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continu…
New capillary Christoffel-Minkowski problem solved for half-space.
problem Finding strictly convex capillary hypersurfaces from capillary functions.
method Established analogous result in capillary setting for half-space.
result Capillary functions lead to strictly convex capillary hypersurfaces.
The space of polygons up to similarity is studied using the Schwarz-Christoffel formula.
problem Understanding the space of polygons up to similarity.
method Using the Schwarz-Christoffel formula to show homeomorphism and proving embeddings for specific polygons.
result The space of labelled simple polygons up to similarity is homeomorphic to R^(2n-4) for n=3,4,5.
This paper optimizes sampling for least-squares approximation.
problem Optimizing sampling for least-squares approximation in arbitrary linear spaces.
method Introducing the Christoffel function to construct near-optimal random sampling strategies.
result The number of samples scales log-linearly in the dimension of the approximation space.
Outlier detection methods have become increasingly relevant in recent years due to increased security concerns and because of its vast application to different fields. Recently, Pauwels and Lasserre (2016) noticed that the sublevel sets of the inverse Christoffel function accurately depict the shape of a cloud of data …
We formulate the Riemannian calculus of the probability set embedded with -Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability m…
Author finds the solutions of the Christoffel problem for open and closed surfaces in Riemannian space. The Christoffel problem is reduced to the problem of construction the continuous G-deformations preserving the sum of principal radii of curvature for every point of surface in Riemannian space. G-deformation transfe…
Quadrics and Scherk towersmath.DG
We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic fu…