The Indefinite Sphere

The ‘Indefinite’ Sphere

I collect here mathematical fragments to probe the symmetries of ‘indefinite spheres’ which are not studied that I know of.

Consider an n dimensional vector space V over the reals, and a non-degenerate symmetric bilinear form f over V, with signature (p, q). f also provides a metric, or pseudo metric δ on V: δ(x, y) = √f(x−y, x−y).

Consider the indefinite sphere S = {x | f(x, x) = 1}. S = ∅ if p=0. We assume p>0. (motivation) Consider too the (pseudo) embedding metric on S derived from δ. But what is the signature of that metric? There is a group of linear transformations on V, called the Indefinite orthogonal group, that leave f unchanged and thus map S onto S. This can be seen to leave the pseudo metric on S unchanged as well. This group is a Lie group with dimension n(n−1)/2 and S is an n−1 dimensional manifold. Thus S is uniformly intrinsically curved. When f is positive definite (signature (n, 0)) the group is the Orthogonal group O(n, ℝ) and S is the ordinary sphere Sⁿ⁻¹.

The topology of S depends on p and q. S is empty for p=0. S is disconnected with two components for p=1. otherwise p−1-connected but not p-connected.

Here is an equivalence relation between symmetric bilinear forms: f is equivalent to f' iff for some invertible linear transformation A on V, for all x and y f'(x, y) = f(Ax, Ay).

Sylvester showed how to choose n vectors e_i in V so that for some integers I, J and K:

0 ≤ I, J, K
I+J+K = n
If 0≤i<I then f(e_i, e_i) = 1
If I+J≤i<n then f(e_i, e_i) = −1
f(e_i, e_j) = 0 otherwise.

f determines I, J and K and (I, J and K determine f to within a linear transformation on V). There are (n+1)(n+2)/2 equivalence classes of quadratic forms. S = ∅ just if I=0; That leaves n²/2 + (3/2)n + 1 − (n+1) = n(n+1)/2 classes of pshpers.

For any basis {e_i} for V there exist n² g’s reals {f_ij} such that
f(Σ_iα_ie_i, Σ_iβ_ie_i) = Σ_ijα_iβ_jf_ij
For a fixed basis f determines the g’s and conversely.
f is symmetric ≜ f(x, y) = f(y, x) ↔︎ g_ij = g_ji.

Henceforth we assume that J=0 which corresponds to a non-degenerate f.

Sylvester showed how to find a basis {e_i} for V where the equation for f is:
f(x, y) = Σ[0≤i<I]x_iy_i − Σ[I≤i<n]x_iy_i.
The equation for S in V is thus f(x, x) = 1 or
Σ[0≤i<I]x_i² − Σ[I≤i<n]x_i² = 1.

For j<I it is convenient to compute the curvature tensor on S at the point Z where e_j = 1 and e_j = 0 for j≠i. At Z, S is tangent to the hyper plane thru V defined by e_j = 1.

except that when I = 0; those equivalence

To be said: For signature (n, 0) we have the sphere and it is n-connected. For signature (p, q) S is p-connected.
Associated with the form f there is the Indefinite orthogonal group of linear transformations on V that preserve f: if A is in that group then f(x, y) = f(Ax, Ay).