# Manifold and metric

Does a manifold necessarily have a metric?
Does a manifold without metric exist? If it exists, what is its name?

HallsofIvy
Homework Helper
Since a manifold is locally Euclidean, it must always have a local metric. However, it does not follow that there will be a "distance" between ANY two points and so there may not be a "global" metric.

quasar987
Homework Helper
Gold Member
the hierarchy is something like this. You take a set. Add a topology. It becomes a tpological space. Add an atlas, it becomes a topological manifold. Change for a differentiable atlas. It becomes a differential manifold. Add a Riemannian or Pseudo-Riemannian of Lorentzian or whatever metric and it becomes a Riemannian (resp. Pseudo-Riemannian, Lorentzian, whatever) manifold.

Topological manifolds (sets with a topology locally homeomorphic to Rn) do not necessarily admit a metric. There are then many non-metrizable manifolds, such as the Prüfer manifold. Urysohn's metrization theorem will let you know if your manifold is metrizable.

Last edited:
Chris Hillman