Lie Sphere Geometry - Lie Sphere Geometry in Space and Higher Dimensions - General Theory

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 Q_n 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 Q_n 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, ∈ Q_n 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.