(in progress)

Since the spaces \mathbb{H}_s are Lie groups, it suffices to analyze their metric properties at the origin. Recall that the geodesic equations are:

x\prime\prime+k y\prime = 0
y\prime\prime - k x\prime = 0
k\prime=0

k = \frac{s}{(1-s)^2}\left(z\prime +\frac{s}{2}(x\prime y - y\prime x)\right)

Solving these starting at the origin and assuming unit speed,

x(t) = a (\cos(kt) -1) + b \sin(kt)

y(t) = a \sin(kt) - b(\cos(kt) - 1)

z(t) = \frac{s}{k}t + \frac{s}{2} \left(a^2+b^2\right) \sin (k t)

For reference,

x\prime(0)=k b

y\prime(0) = k a

z\prime(0) = \frac{s}{k} + k \frac{s}{2}(a^2+b^2)

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Last Updated ( Monday, 25 May 2009 13:40 )