z1 = ρλ1, z2 = ρλ2, ... zn = ρλn,
we obtain
| ds² = Σdz² + | 1 | { ( | R | sinh | ρ | ) | ² | − 1 } Σ (zidzk − zkdzi)². |
| ρ² | ρ | R |
Hence when ρ is small, we have approximately
| ds² = Σdz² + | 1 | Σ (zidzk − zkdzi)² |
| 3R² |
(1).
Considering a surface element through the origin, we may choose our axes so that, for this element,
z3 = z4 = ... = zn = 0.
Thus
| dz1² + dz2² + | 1 | (z1dz2 − z2dz1)² |
| 3R² |