Differentiable manifold not riemannian

Strictly speaking, "Riemannian" implies the metric is positive definite, so you can just use a metric with indefinite signature...

Or, you can simply not specify a metric. Voila! It's easy to define differentiable manifolds without metrics...you just don't give them a metric.

Or do you want an example of a differentiable manifold that is not metrizable? That might be more difficult. I can't think of any examples.

Edit: In fact, there can be no examples. Every differentiable manifold can be embedded in R^n for some n, and therefore can always be given a metric by taking the induced metric from R^n.

 

It depends on your definition of differentiable manifold. If you only have that it is locally homeomorphic to Euclidean space and the overlapping coordinates give a differentiable function, then you can have weird things like the long line, which is defined as:
Pick an uncountable ordinal W. Take the set [0,1)xW (an uncountable number of copies of [0,1). This is essentially too long to be embedded into Euclidean space. I imagine it's not metrizable because if two copies of [0,1) are infinitely far apart the distance between them probably has to be infinite, but I can't think of a reason why so don't take that as fact

 

The proof that every manifold has a metric (as well as the proof of Whitney's embedding theorem) relies on paracompactness. If you drop this requirement, you can have all sorts of aberrations.

In fact, if your space has a metric, it has to be second countable (delta-balls type argument).

 

We have to be careful with definitions here. In particular, it seems to me that a non-second-countable smooth manifold may be equipped with a local Riemann metric but not be a metric space.

 

True, true. The space must be path-connected for what I said to hold.

 

Not enough. If it's non-second-countable, the integral that lets us go from local Riemannian metric to global metricity may diverge.

 

I don't follow. |c'(t)| is a continuous function on a compact set. How could its interval diverge?

 

Nevermind, I was wrong.

 

The real two dimensional vector space R².