Projective background of the infinitesimal rigidity of frameworks

Izmestiev, Ivan

doi:10.1007/s10711-008-9339-9

Cited by 32 publications

(44 citation statements)

References 31 publications

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“…In the Riemannian case these results are contained inside a general statement saying that each times that there is a map from a manifold to a flat manifold sending geodesics to geodesics then there exists a (unique) map of the associated tangent bundles sending Killing fields to Killing fields [Vol74] (this result should be checked for pseudo-Riemannian context). Concerning the polyhedral surfaces, there exists a more geometric way to define the infinitesimal Pogorelov maps [SW07,Izm08].…”

Section: Fuchsian Infinitesimal Rigiditymentioning

confidence: 99%

Fuchsian polyhedra in Lorentzian space-forms

Fillastre

2010

Math. Ann.

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Abstract. Let S be a compact surface of genus > 1, and g be a metric on S of constant curvature K ∈ {−1, 0, 1} with conical singularities of negative singular curvature. When K = 1 we add the condition that the lengths of the contractible geodesics are > 2π. We prove that there exists a convex polyhedral surface P in the Lorentzian space-form of curvature K and a group G of isometries of this space such that the induced metric on the quotient P/G is isometric to (S, g). Moreover, the pair (P, G) is unique (up to global isometries) among a particular class of convex polyhedra, namely Fuchsian polyhedra.This extends theorems of A.D. Alexandrov and Rivin-Hodgson [Ale42, RH93] concerning the sphere to the higher genus cases, and it is also the polyhedral version of a theorem of Labourie-Schlenker [LS00].Math. classification: 52B70(52A15,53C24,53C45)

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Section: Fuchsian Infinitesimal Rigiditymentioning

confidence: 99%

Fuchsian polyhedra in Lorentzian space-forms

Fillastre

2010

Math. Ann.

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“…It is known that the Möbius geometry of R 3 ∪ {∞} is a subgeometry of the projective geometry of RP 4 . Möbius transformations of R 3 ∪{∞} are represented as projective transformations of RP 4 preserving the quadric defined by the light cone L. If two non-degenerate realizations are related by a projective transformation, then the spaces of self-stresses of the two realizations are isomorphic [23]. Hence, we obtain another proof of Theorem 1.3.…”

Section: Möbius Invariancementioning

confidence: 85%

“…Then the above corollary is simply a special case of the projective invariance of infinitesimal rigidity [23]. In the smooth theory a minimal surface is the Christoffel dual of its Gauß map.…”

Section: Example: Inscribed Triangular Meshesmentioning

confidence: 97%

Isothermic triangulated surfaces

Lam

Pinkall

2016

Math. Ann.

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Abstract. We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes.A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.

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“…The equivalence relations from Definition 3.1 ensure that a linear extension is well-defined. For a proof of its bijectivity, see [24].…”

Section: Equivalence Of Static and Infinitesimal Rigiditymentioning

confidence: 99%