Results 1 to 10 of about 2,719,579 (161)
Another approach to the fundamental theorem of Riemannian geometry in
Journal de Mathématiques Pures et Appliquées, 2006 AbstractIn 1992, C. Vallée showed that the metric tensor field C=∇ΘT∇Θ associated with a smooth enough immersion Θ:Ω→R3 defined over an open set Ω⊂R3 necessarily satisfies the compatibility relationCURLΛ+COFΛ=0in Ω, where the matrix field Λ is defined in terms of the field U=C1/2 byΛ=1detU{U(CURLU)TU−12(tr[U(CURLU)T])U}.The main objective of this paper
Philippe G. Ciarlet +4 more
semanticscholar +8 more sources
Riemannian Geometry and the Fundamental Theorem of Algebra [PDF]
arXiv, 2011 If a (non-constant) polynomial has no zero, then a certain Riemannian metric is constructed on the two dimensional sphere. Several geometric arguments are then shown to contradict this fact.
J. M. Almira, Alfonso Romero
arxiv +5 more sources
The Reciprocal of the Fundamental Theorem of Riemannian Geometry [PDF]
arXiv, 2009 The fundamental theorem of Riemannian geometry is inverted for analytic Christoffel symbols. The inversion formula, henceforth dubbed Ricardo's formula, is obtained without ancillary assumptions. Even though Ricardo's formula can mathematically give the full answer, it is argued that the solution should be taken only up to a constant conformal factor ...
Héctor H. Calderón
arxiv +5 more sources
Warped Riemannian metrics for location-scale models [PDF]
arXiv, 2017 The present paper shows that warped Riemannian metrics, a class of Riemannian metrics which play a prominent role in Riemannian geometry, are also of fundamental importance in information geometry.
A Terras +42 more
core +4 more sources
Riemannian Structures on
Mathematics, 2020 Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as ...
Andrew James Bruce, Janusz Grabowski
doaj +4 more sources
Rotation fields and the fundamental theorem of Riemannian geometry in
Comptes Rendus. Mathématique, 2006 Let Ω be a simply-connected open subset of R^3. We show that, if a smooth enough field U of symmetric and positive-definite matrices of order three satisfies the compatibility relation (due to C. Vallee) CURL Λ+COF Λ=0 in Ω, where the matrix field Λ is defined in terms of the field U by Λ=(1/detU){U(CURL U)^T U−(1/2)(tr[U(CURL U)T])U ...
Philippe G. Ciarlet +4 more
semanticscholar +5 more sources
On the Non Metrizability of Berwald Finsler Spacetimes [PDF]
Universe, 2020 We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund ...
Andrea Fuster +3 more
doaj +4 more sources
Nonlinear elasticity complex and a finite element diagram chase [PDF]
arXiv.org, 2023 In this paper, we present a nonlinear version of the linear elasticity (Calabi, Kr\"oner, Riemannian deformation) complex which encodes isometric embedding, metric, curvature and the Bianchi identity. We reformulate the rigidity theorem and a fundamental
Hu, Kaibo
core +2 more sources
Massively parallel computation of globally optimal shortest paths with curvature penalization
Concurrency and Computation: Practice and Experience, Volume 35, Issue 2, 25 January 2023., 2023 Abstract We address the computation of paths globally minimizing an energy involving their curvature, with given endpoints and tangents at these endpoints, according to models known as the Reeds‐Shepp car (reversible and forward variants), the Euler‐Mumford elasticae, and the Dubins car. For that purpose, we numerically solve degenerate variants of the
Jean‐Marie Mirebeau +4 more
wiley +1 more source
Riemannian curvature measures [PDF]
Geometric and Functional Analysis, 2017 A famous theorem of Weyl states that if M is a compact submanifold of euclidean space, then the volumes of small tubes about M are given by a polynomial in the radius r, with coefficients that are expressible as integrals of certain scalar invariants of ...
Joseph H. G. Fu, Thomas Wannerer
semanticscholar +2 more sources