Results 1 to 10 of about 872 (123)
Another approach to the fundamental theorem of Riemannian geometry in
Journal de Mathématiques Pures et Appliquées, 2007 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
Ciarlet, Philippe +4 more
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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 ...
Ciarlet, Philippe G. +4 more
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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
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Riemannian Geometry and the Fundamental Theorem of Algebra
, 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.
Almira, J. M., Romero, A.
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The Reciprocal of the Fundamental Theorem of Riemannian Geometry
, 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
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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
Kaibo Hu
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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
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On the Non Metrizability of Berwald Finsler Spacetimes
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
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Hierarchy structures in finite index CMC surfaces [PDF]
Advances in Calculus of Variations, 2022 Given ε 0 > 0 {{\varepsilon}_{0}>0} , I ∈ ℕ ∪ { 0 } {I\in\mathbb{N}\cup\{0\}} and K 0 , H 0 ≥ 0 {K_{0},H_{0}\geq 0} , let X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ ε 0 {\operatorname{Inj}(X)\geq{\varepsilon}_{0}} and ...
William H. Meeks III, Joaquín Pérez
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Warped Riemannian Metrics for Location-Scale Models [PDF]
Geometric Structures of Information, 2017 The present contribution 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.
S. Said, L. Bombrun, Y. Berthoumieu
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