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Geodesic trapped in one half of the unit one-sheet hyperboloid

For a unit-speed geodesic on

x^{2} + y^{2} = z^{2} + 1

, Clairaut's constant

c

gives

\overset{z}{˙}^{2} = \frac{z ^{2} + 1 - c ^{2}}{1 + 2 z ^{2}} .

If

c > 1

, the geodesic has a turning point at

z = c^{2} - 1

and remains entirely in one of the regions

z > 0

or

z < 0

.

 Ancestors (7)

 Incoming links (1)

Solution