Theorems · Theorem · number theory
UpperHalfPlane.gl_smul_eq_self_iff_dist_sq_eq
∀ {g : GL (Fin 2) ℝ} {z : UpperHalfPlane},
(↑g).det < 0 →
(↑g).trace = 0 → ↑g 1 0 ≠ 0 → (g • z = z ↔ dist (↑z) (-↑(↑g 1 1) / ↑(↑g 1 0)) ^ 2 = -(↑g).det / ↑g 1 0 ^ 2)If g is an orientation reversing matrix with trace zero and c ≠ 0,
then its action on the upper half plane has a half-circle of fixed points.
In the hyperbolic geometry, this half-circle is a line.
If c = 0, then this line is a vertical half-line in the usual geometry,
see gl_smul_eq_self_iff_re_eq.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound
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- UpperHalfPlane.gl_smul_eq_self_iff_dist_eqproof · cited by 1