gregeganSF, 7 months ago

The standard metric on the sphere in polar coordinates is:

dθ^2 + sin^2(θ) dφ^2

But what if we change this to:

(1 + h(cos(θ))^2 dθ^2 + sin^2(θ) dφ^2

for some function h(x) from [-1,1] that satisfies:

h(-x) = -h(x)
|h(x)| < 1
h(1)=0

It might seem that we’ve just distorted the sphere arbitrarily, but requiring h to be an odd function means several global quantities remain unchanged.

Along geodesics of the new surface, we have:

dφ/dθ = sin θ_0 (1 + h(cos θ)) / (sin θ √[sin^2(θ) - sin^2(θ_0)])

where the geodesic is tangent to circles of latitude at θ_0 and π-θ_0. But if we integrate this between those endpoints, because h is odd we always get:

Δφ = π

So, the longitude of every geodesic changes by π as it moves between its two extremes of latitude.

Similarly, for the distance s along a geodesic:

ds/dθ = sin θ (1 + h(cos θ)) / √[sin^2(θ) - sin^2(θ_0)]

with an integral between the same circles of latitude:

Δs = π

Joining this curve to its mirror image always gives a closed curve of length 2π.

We also get an unchanged total surface area of 4π.

Reference: “Manifolds all of whose geodesics are closed” by A L Besse, Ch 4. H/T @RobJLow

A sphere deforms, while a set of geodesics on it continue to be closed loops