ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract We give here new results of topology and integral geometry concerning the Gauss linking number I of closed manifolds inn-dimensional space. The rigid manifolds have arbitrary shapes and dimensions, and are statistically at random positions in ℝ n . Generalizing Pohl's work, for two closed manifoldsC 1 r ,C 2 s , of respective dimensionsr ands, with 0≦r≦n−1, andr+s+1=n, we consider the “kinematic linking integral”I=〈∝I 2(x,O)d n x〉, of the square linking number I ofC 1 r andC 2 s , over the group of Euclidean motions of one manifold (translationsx, rotationsO). Introducing a new tensorial method, and using group theory, we show quite generally thatI=num. fact. $$\int\limits_0^\infty {d\varrho [\mathcal{A}_1 (\varrho )\mathcal{A}_2 (\varrho ) + \delta _{r,s} \mathcal{B}_1 (\varrho )\mathcal{B}_2 (\varrho )]} $$ , where ϱ is a length variable and whereA α,ℬα ℬα(α=1, 2) are characteristic functions associated with the manifoldC α only. We study functionsA and ℬ of a manifoldC r , of dimensionr, in all cases 0≦r≦n−1.A always exists.A(0) givesC's area, whereas $$\int\limits_0^\infty {\mathcal{A}(\varrho )d\varrho } $$ equals the interior volume of a hypersurfaceC. ℬ is found to exist and not to vanish only if 2 dimC+1=n andn=3+4q=3, 7, 11 ...A and ℬ are explicitly calculated for segments andr-spheresS r . As an application the topological excluded volume of a gas of nonlinked spheresS r moving in ℝ2r+1 is calculated. We generalize toN manifoldsC α, α=1, ...,N, linked successively to each other and forming a ring. The cyclic product of their linking numbers is integrated over the group of motions of the manifolds. It is shown to factorize completely in Fourier space, with special algebraic rules, over the set of 2N characteristic functionsA α, ℬα, associated with theC α's. The same algebra of characteristic functions is shown to describe a larger class of topology and electromagnetism properties: a new theorem is given for a family of Euclidean group integrals involving the random linking numbers, mutual inductances and contact distributions ofN manifolds.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01254458
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