Generalization of theorems of Darboux-Sauer and Pogorelov

Volkov, Yu. A.

doi:10.1007/bf01476850

Cited by 5 publications

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“…Other proofs can be found in Wunderlich [39] for frameworks, and Volkov [31] for smooth submanifolds of the Euclidean space.…”

Section: Lemmamentioning

confidence: 99%

“…Volkov in [31] gives a unified treatment of Darboux-Sauer correspondence and infinitesimal Pogorelov maps for smooth surfaces.…”

Section: Remarksmentioning

confidence: 99%

“…In the smooth case, Volkov [Vol74] proves that a map between Riemannian manifolds that sends geodesics to geodesics maps infinitesimally flexible hypersurfaces to infinitesimally flexible ones. Since projective maps of E d to itself and gnomonic projections of the Euclidean space to the spherical and hyperbolic spaces send geodesics to geodesics, Volkov's theorem includes Darboux' and Pogorelov's as special cases.…”

Section: Remarksmentioning

confidence: 99%

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Projective background of the infinitesimal rigidity of frameworks

Izmestiev

2008

Geom Dedicata

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We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the second one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality between statics and kinematics is established through the principles of virtual work. A geometric approach to statics, due essentially to Grassmann, makes both theorems straightforward. Besides, it provides a simple derivation of the formulas both for the Darboux-Sauer correspondence and for the infinitesimal Pogorelov maps.

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“…Other proofs can be found in Wunderlich [39] for frameworks, and Volkov [31] for smooth submanifolds of the Euclidean space.…”

Section: Lemmamentioning

confidence: 99%

“…Volkov in [31] gives a unified treatment of Darboux-Sauer correspondence and infinitesimal Pogorelov maps for smooth surfaces.…”

Section: Remarksmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

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Projective background of the infinitesimal rigidity of frameworks

Izmestiev

2008

Geom Dedicata

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“…126,127 in [21] for the relations between infinitesimal affine transformations and Killing fields. • Lemma 4.18 was proved in [20] and independently in [46]. See also the end of Section 5.5.…”

Section: Comments and Referencesmentioning

confidence: 87%

“…• Volkov [46] showed the following: a vector field Z on S is an infinitesimal isometric deformation if and only if for each x ∈ S there exists a Killing field K x such that Zx = (K x )x and (∇X Z)x = (∇X K x )x ∀X ∈ TxS .…”

Section: Comments and Referencesmentioning

confidence: 99%

Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions

Fillastre

Seppi

2019

IRMA Lectures in Mathematics and Theoretical Physics

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3 The notion of Levi-Civita connection is recalled in Section 4. 4 Of course, if the tensor is not positive definite, it does not induce a distance on M , and in particular there is no notion of "minimization of distance" for the geodesics. 5 A smooth manifold is pseudo-Riemannian if it is endowed with a non-degenerate smooth (0, 2)-tensor. If the tensor is positive definite at each point, then the manifold is Riemannian. If the tensor has a negative direction, but no more than one linearly independent negative direction at each point, then the manifold is Lorentzian. 6 The isometry group of M is bigger than O(p, q) if M is not connected. 7 This follows from the Gauss formula and the fact that the shape operator on M is ± the identity. The fact that the sectional curvature is constant also follows from the preceding item.It is obvious that the projective quotient of M is anti-isometric to M, and will thus be denoted by M. Moreover, in the case where M is not connected, it has two connected components, which correspond under the antipodal map x → −x. In particular, M is connected.By construction, M is a subset of the projective space RP n . Indeed, topologically M can be defined asor the same definition with > replaced by <, depending on the case. Hence any choice of an affine hyperplane in R n+1 which does not contain the origin will give an affine chart of the projective space, and the image of M in the affine chart will be an open subset of an affine space of dimension n.Recall that PGL(n + 1, R), the group of projective transformations (or homographies), is the quotient of GL(n + 1, R) by the non-zero scalar transformations. Even if M is not connected, an isometry of M passes to the quotient only if it is an element of O(p, q). Hence Isom(M), the isometry group of M, is PO(p, q), the quotient of O(p, q) by {±Id} (indeed elements of O(p, q) have determinant equal to 1 or −1). Lines and pseudo-distancePseudo-distance on M. A geodesic c of M is the non-empty intersection of M with a linear plane. This intersection might not be connected. The geodesic c of M is said to be

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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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Generalization of theorems of Darboux-Sauer and Pogorelov

Cited by 5 publications

References 1 publication

Projective background of the infinitesimal rigidity of frameworks

Projective background of the infinitesimal rigidity of frameworks

Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions

Fuchsian polyhedra in Lorentzian space-forms

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