# Finite Metric Spaces

Is every finite metric space imbeddable in a manifold?

That is, for every finite metric space (X,d), does there exist some manifold such that there are |X| many points on it which is isometric (with the length of geodesics as metric) to (X,d)?

Can you do a set with 4 points?

What do you mean?

Hurkyl
Staff Emeritus
Gold Member
"Every finite metric space" is a tall order. Try starting with embedding "every metric space with 4 points" into Riemannian manifolds. (Actually, why not start with 0, 1, 2, 3, and then 4?)

(P.S. I assume you mean "connected and Riemannian" manifolds?)

OK, it seems this one

o
\
o ---- o
/
o

where the metric is the number of steps between nodes, can't be done. But I can't prove it conclusively.