Gravity

A Short Course in General Relativity, Edition: 2nd ed by J. Foster, J. Nightingale

By J. Foster, J. Nightingale

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51), into an isometric polyhedral surface (Tl =a , M) which is no longer equilateral. 53) (Tl , M) −→ Fl(Tl ,M) , that associates to (Tl , M) its orbit under a generic finite sequence, Fl T , T (Tl , M), of isometric flip moves met (M)| (Tl , M) = Fl T , T (Tl , M) . 54) Fl(Tl ,M) is the singular Euclidean structure Mpol associated with the polyhedral surface (Tl , M). 55) Fl(T l ,M) . 24 1 Triangulated Surfaces and Polyhedral Structures Conversely, let Mpol ∈ POL g,N0 (M) a singular Euclidean structure and let met (M).

E. a suitable notion of tangent space to Tg,N l 0 ize the intuitive picture of what a deformation of a neighborhood of a vertex σ 0 (k) in a polyhedral surface looks like. To this end, we inject the given vertex σ 0 (k) in the origin O of (R3 ; O, x, y, z), where we identify R3 , endowed with the vector product ×, with the Lie algebra so(3) of the rotation group SO(3). Since σ 0 (k) is a conical point whose generators are edges of Euclidean triangles, we can associate with σ 0 (k) the polyhedral cone with apex O, whose generators have lengths normalized to 1 and whose directrix is a piecewise-geodesic curve on the surface of a unit sphere S2 centered at the origin.

This length is the conical angle Θ(k) at the vertex. 77). 77) also induces on the curve c(k) the decoration defined by the q(k) points {pα } images of the vertices {σ 0 (hα )} ∈ link(k). 78) of the arc of geodesic c(k) (α, α + 1) between the point pα and pα+1 . 6 The {c(k) (α, α + 1)} are geodesic arcs since they are defined by the intersection between S2 and planes passing through the center of S2 . Kapovich and Millson have studied in depth the properties of the set of spherical polygons and of the associated moduli spaces, (see [9] and references therein).

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