- 117

- 0

## Main Question or Discussion Point

I'm looking for a simple example of a differentiable manifold that doesn't have an associated riemann metric.

thanks

thanks

- Thread starter redrzewski
- Start date

- 117

- 0

I'm looking for a simple example of a differentiable manifold that doesn't have an associated riemann metric.

thanks

thanks

Ben Niehoff

Science Advisor

Gold Member

- 1,864

- 160

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

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.

Office_Shredder

Staff Emeritus

Science Advisor

Gold Member

- 3,734

- 99

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

- 489

- 0

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

- 906

- 2

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.

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

- 489

- 0

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

- 906

- 2

Not enough. If it's non-second-countable, the integral that lets us go from local Riemannian metric to global metricity may diverge.True, true. The space must be path-connected for what I said to hold.

- 489

- 0

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

- 906

- 2

Nevermind, I was wrong.

dx

Homework Helper

Gold Member

- 2,003

- 18

The real two dimensional vector space R^{2}.

- 117

- 0

His definition of differentiable manifold looks to be as OfficeShredder says. Arnold assumes it is connected as well, but there doesn't appear to be the 2nd countable requirement (that Lee explicitly calls out for instance).

I was confused since Arnold calls out adding the additional structure of the riemann metric.

Thanks for all the clarifications.

- Last Post

- Replies
- 1

- Views
- 2K

- Last Post

- Replies
- 16

- Views
- 5K

- Last Post

- Replies
- 2

- Views
- 3K

- Last Post

- Replies
- 0

- Views
- 2K

- Last Post

- Replies
- 5

- Views
- 1K

- Last Post

- Replies
- 22

- Views
- 2K

- Replies
- 4

- Views
- 3K

- Replies
- 3

- Views
- 1K