General Theory
Lie sphere geometry in n-dimensions is obtained by replacing R3,2 (corresponding to the Lie quadric in n = 2 dimensions) by Rn + 1, 2. This is Rn + 3 equipped with the symmetric bilinear form
The Lie quadric Qn is again defined as the set of ∈ RPn+2 = P(Rn+1,2) with x · x = 0. The quadric parameterizes oriented (n – 1)-spheres in n-dimensional space, including hyperplanes and point spheres as limiting cases. Note that Qn is an (n + 1)-dimensional manifold (spheres are parameterized by their center and radius).
The incidence relation carries over without change: the spheres corresponding to points, ∈ Qn have oriented first order contact if and only if x · y = 0. The group of Lie transformations is now O(n + 1, 2) and the Lie transformations preserve incidence of Lie cycles.
The space of contact elements is a (2n – 1)-dimensional contact manifold Z2n – 1: in terms of the given choice of point spheres, these contact elements correspond to pairs consisting of a point in n-dimensional space (which may be the point at infinity) together with an oriented hyperplane passing through that point. The space Z2n – 1 is therefore isomorphic to the projectivized cotangent bundle of the n-sphere. This identification is not invariant under Lie transformations: in Lie invariant terms, Z2n – 1 is the space of (projective) lines on the Lie quadric.
Any immersed oriented hypersurface in n-dimensional space has a contact lift to Z2n – 1 determined by its oriented tangent spaces. There is no longer a preferred Lie cycle associated to each point: instead, there are n – 1 such cycles, corresponding to the curvature spheres in Euclidean geometry.
The problem of Apollonius has a natural generalization involving n + 1 hyperspheres in n dimensions.
Read more about this topic: Lie Sphere Geometry, Lie Sphere Geometry in Space and Higher Dimensions
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