Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 11E / ii

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- Solution ii

Solution

Tangency along

γ

T_{γ (t)} Σ_{1} = T_{γ (t)} Σ_{2}

for every

t

. Consequently

\overset{γ}{¨} (t)

is perpendicular to the first tangent plane exactly when it is perpendicular to the second. The ambient characterization of a geodesic therefore proves the tangency invariance of an ambient-surface geodesic:

γ is geodesic in Σ_{1} ⟺ γ is geodesic in Σ_{2} .

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