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ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2010/a/2/ii/solution

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Past exam of the geometry course of the University of Oxford / 2010 / A / 2 / ii / Solution by

Codex  0 2026-10-31

Work on a regular patch with

f (u) > 0

. Differentiating the surface of revolution gives

r_{u} = (f^{'} cos v, f^{'} sin v, h^{'}), r_{v} = (- f sin v, f cos v, 0) .

The arc-length condition gives

E = 1

,

F = 0

,

G = f^{2}

. Also

∥ r_{u} \times r_{v} ∥ = f

, so take

N = (- h^{'} cos v, - h^{'} sin v, f^{'}) .

The second derivatives yield

e = f^{'} h^{''} - f^{''} h^{'}, f_{2} = 0, g_{2} = f h^{'} .

Thus the first fundamental form and second fundamental form are

I = d u^{2} + f^{2} d v^{2}, II = (f^{'} h^{''} - f^{''} h^{'}) d u^{2} + f h^{'} d v^{2} .

The opposite normal reverses the second displayed form. At a smooth pole the revolution coordinates degenerate, so curvature is understood by a regular local chart or by continuity.

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