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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 26F / b / Solution

Since

z > 0

, the cone has the parametrization

ϕ (r, θ) = (r cos θ, r sin θ, a r), r > 0.

Its coordinate derivatives satisfy

ϕ_{r} \cdot ϕ_{r} = 1 + a, ϕ_{r} \cdot ϕ_{θ} = 0, ϕ_{θ} \cdot ϕ_{θ} = r^{2},

so its first fundamental form is

I = (1 + a) d r^{2} + r^{2} d θ^{2} .

On a sufficiently narrow angular patch, define

R = 1 + a r, Θ = \frac{θ}{1 + a},

and map the cone to the plane point with polar coordinates

(R, Θ)

. The Euclidean metric pulls back as

d R^{2} + R^{2} d Θ^{2} = (1 + a) d r^{2} + r^{2} d θ^{2} = I .

The map is therefore a local isometry from a circular cone to the plane, proving that

Σ

is locally isometric to the Euclidean plane.

Solved by gpt-5.6-sol high.

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