By A. Libgober, P. Wagreich

ISBN-10: 3540108335

ISBN-13: 9783540108337

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Read e-book online Discrete Differential Geometry PDF

Discrete differential geometry is an energetic mathematical terrain the place differential geometry and discrete geometry meet and engage. It offers discrete equivalents of the geometric notions and strategies of differential geometry, akin to notions of curvature and integrability for polyhedral surfaces.

Extra resources for Algebraic Geometry. Proc. conf. Chicago, 1980

Example text

Where Γ is a lifted curve and f denotes df /dx. It follows that |Γ, Γx , . . x | = |Γ, Γy , . . y |(f )n(n+1)/2 , and, therefore, the Wronski determinant W (x) is a tensor density of degree n(n + 1)/2, that is, an element of Fn(n+1)/2 . 5). 4) are −n/2-densities. 3) it follows that the kernel of the operator A consists of −n/2-densities. It remains to note that the kernel uniquely defines the corresponding operator. The brevity of the proof might be misleading. 3 by a direct computation. g.

We define a canonical PGL(2, R)-isomorphism between this space and the space of tensor densities on S 1 . We apply this isomorphism to construct projective differential invariants of non-degenerate curves in RPn . Space of symbols There is a natural filtration 0 1 k Dλ,µ (S 1 ) ⊂ Dλ,µ (S 1 ) ⊂ · · · ⊂ Dλ,µ (S 1 ) ⊂ · · · The corresponding graded Diff(S 1 )-module Sλ,µ (S 1 ) = gr (Dλ,µ (S 1 )) is called the module of symbols of differential operators. k (S 1 )/D k−1 (S 1 ) is isomorphic to the module of The quotient-module Dλ,µ λ,µ tensor densities Fµ−λ−k (S 1 ); the isomorphism is provided by the principal 46 CHAPTER 2.

Indeed, fix a parameter value x = x 0 , (j) and choose a special basis φ0 , . . , φn ∈ U such that φi (x0 ) = 0 for all (i) i = j; i, j = 0, . . , n, and φi (x0 ) = 1 for all i. Let ψi ∈ V be the basis in V defined similarly. In these bases, the matrix of B(φ, ψ)(x 0 ) is triangular with the diagonal elements equal to ±1. ˜ The pairing B allows us to identify U ∗ with V . 6). Let Γ(x) ⊂ V be a similar curve corresponding to A ∗ . We want to show that these two curves are dual with respect to the pairing B, that is, ˜ (i) (x0 ), Γ(x0 )) = 0, i = 0, .