Theorem. On the Heisenberg group
Z 1. Geodesics are horizontal lifts of Euclidean lines and circles in the plane. heisenberg copy 2. Metric lines are horizontal lifts of Euclidean lines in the plane. Y x 3 as a manifold: R as a metric space: x Call a smooth path c(t) = (x(t), y(t),dyed z(t)) ``horizontal’’ if : dz 1 dx dy = (x(t) O y(t) O) dt 2 dt dt and call the length of such a path : b dx 2 dy 2 `(c):= dx2 + dy2 := + dt c a s dt dt Z p Z Then d(A, B) = inf { length of c: c horizontal, c joins A to B }
as a Lie group: (x,y,z)(x’, y’, z’) = (x + x’, y+ y’, z+ z’) + (0,0, (1/2)(xy’ - yx’)) d is a left-invariant metric on the Heisenberg group: d(gA, gB) = d(A, B)
The projection 3 2 ⇡ : R R ; ⇡(x, y, z)=(x, y) ! is a submetry:
DEF: a submetry is an onto map F between metric spaces such that F(B(p, r)) = B(F (p ), r)
`(c)=` 2 (⇡(c)) = ⇡ is a submetry R )
center 0,921 horn spaces t f I c h8 IT
r R2
c is the horizontal lift of