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Higher-dimensional linking integrals. (English) Zbl 1221.57038
A hypersurface \(M\) in Euclidean space \({\mathbb R}^n\) is said to be visible if every ray from the origin either misses \(M\) or meets it just once transversally. For example the unit \(n\)-sphere, \(S^n\), is visible. The authors derive an integral formula for the linking number, \(Lk(K,L)\) of two closed, oriented, null-homologous submanifolds \(K\) and \(L\) of a visible hypersurface \(M\) in \({\mathbb R}^n\) such that \(\dim(K) + \dim(L) = n-1\). When \(M\) is taken to be \(S^n\), this formula agrees with the formulas obtained independently and by other methods by D. DeTurck and H. Gluck [Mat. Contemp. 34, 233–249 (2008; Zbl 1195.57056)] and by G. Kuperberg [Geom. Funct. Anal. 18, No. 3, 870–892 (2008; Zbl 1169.52004)].

MSC:
57R45 Singularities of differentiable mappings in differential topology
53C20 Global Riemannian geometry, including pinching
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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[1] Dennis DeTurck and Herman Gluck, Electrodynamics and the Gauss linking integral on the 3-sphere and in hyperbolic 3-space, J. Math. Phys. 49 (2008), no. 2, 023504, 35. · Zbl 1153.81348
[2] D. DeTurck and H. Gluck, Linking integrals in the \?-sphere, Mat. Contemp. 34 (2008), 239 – 249. · Zbl 1195.57056
[3] Moritz Epple, Orbits of asteroids, a braid, and the first link invariant, Math. Intelligencer 20 (1998), no. 1, 45 – 52. · Zbl 0918.01014
[4] Carl Friedrich Gauss, Integral formula for linking number, Zur mathematischen theorie der electrodynamische wirkungen (Collected Works, Vol. 5), Koniglichen Gesellschaft des Wissenschaften, Göttingen, 2nd ed., 1833, p. 605.
[5] Greg Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), no. 3, 870 – 892. · Zbl 1169.52004
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