 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-2/38d/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For

g_{θθ} = 1, g_{ϕϕ} = sin^{2} θ,

the nonzero Christoffel symbols needed here are

Γ_{ϕϕ}^{θ} = - sin θ cos θ, Γ_{θϕ}^{ϕ} = Γ_{ϕθ}^{ϕ} = cot θ .

On a line of constant longitude,

\dot{ϕ} = 0

, and the geodesic equation reduces to

\ddot{θ} = 0

after choosing an affine parameter. Thus every meridian is a geodesic.

For a nonconstant line of constant latitude,

\dot{θ} = 0

and the

θ

equation requires

- sin θ cos θ \dot{ϕ}^{2} = 0.

Away from the coordinate-degenerate poles, this holds only when

θ = π /2,

so only the equator is a constant-latitude geodesic.

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