It could of been worded better…but I am assuming you didn’t catch I was referencing the unit sphere. I deleted C[sub]p[/sub] and other math notation to avoid losing viewers.
The generalized diameter is the greatest distance between any two points on the boundary of a closed figure.
In Euclidean space, the shortest distance between two antipodal points on the unit sphere is 2; as expected.
If one naively copies that generalized definition of diameter onto Riemannian geometry:
The shortest distance between two points on the surface of a sphere, as measured along the surface of the sphere, an orthodrome or a segment of a great circle. The shortest distance between two antipodal points on the unit sphere is π (Pi).
As this example only depended on Differential Geometry I expect most math fans probably picked up on my admitted opaque context. Define your terms and check your assumptions when switching domains is the point.
Edit to add for other responses: Note due to p = 2 being the only L[sub]p[/sub] group with SO(3) our value of pi 3.14… is the lowest value it has as p > 2 loses rotational invariance.
Why are you measuring across the sphere in Euclidean geometry, but along the sphere in Riemannian geometry? You can measure in either way in either geometry. And the Riemannian result approaches the Euclidean one, in the limit where the size of the sphere is much less than the radius of curvature.
Because in that case the chord metric is not an intrinsic metric or length metric. In this case the induced intrinsic metric uses the great-circle distance. Sure you could respond with the chordal distance when someone asks you how far it is from NYC to London but until someone develops low cost robotic tunneling machines it doesn’t do them very good to travel between the two. Chordal distance or Euclidean Distance is little value to travel on that curved surface either.
Note the generalized definition I linked to.
While the Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. That Euclidean metric is not assumed in non-Euclidean geometry.
Obviously math proofs are not offered in plain English so you can find some wording mistake to pick at, but why do you assume that a geometry that is not Euclidean will follow Euclidean rules? Heck even the taxicab metric in Euclidean space would be an example were you may use different definitions of “distance”…which is what a metric is.
I’m assuming that by “Riemannian geometry” you mean “Riemannian geometry”, and that by “pi” you mean “pi”, and so on. The problem is not that I’m not understanding what you’re saying in words; the problem is that what you are saying in words is just plain wrong.
I guess he is saying that if you stand somewhere on a positively curved surface, measure out a small circle, and divide the circumference by the diameter, you will obtain a number ever so slightly less than “pi”. Which is true…
On a curved surface, you would not expect the circumference divided by the diameter to be constant for all circles, big and small and with different centers, nor the angles of a triangle to add up to pi, etc. To recover a constant you would have to take the limit, which is basically the Euclidean case.
I think the problem is terminology. Riemannian geometry describes a class of different geometries, where the metric’s behaviour is restricted in a certain way. Euclidean geometry and spherical geometry are both examples of Riemnnian geometries. Also when you say circle, what you mean specifically a circle with circumference 2pi*r.