# Euclidian and Riemann space

1. Dec 26, 2013

### LagrangeEuler

What's the difference between Euclidean and Riemann space? As far as I know $\mathbb{R}^n$ is Euclidean space.

2. Dec 26, 2013

### ShayanJ

Let's replace the word "space" with "manifold" because its more general.
A Riemannian manifold is a manifold having a positive definite metric.
A Euclidean manifold is a special case of a Riemannian manifold where the positive definite metric is a Euclidean metric i.e. $d(x,y)=\sqrt{\sum_i (x_i-y_i)^2}$.

3. Dec 27, 2013

### LagrangeEuler

Tnx. But what other metrics do you have to be positive definite in $\mathbb{R}^n$? According to this is Riemann space also Hilbert space?

Last edited: Dec 27, 2013
4. Dec 27, 2013

### ShayanJ

Anything anyone can think of!!!
For example the taxicab metric.

About your second question,the semi-definite metric making our manifold a Riemannian one,maybe not induced by an inner product!
Also the metric space in question maybe not complete.
So no,not all Riemannian manifolds are Hilbert Spaces!
But it seems to me that every Real Hilbert Space,is a Riemmanian manifold!
(Sorry math people for putting my feet into your shoes!)

Last edited: Dec 27, 2013
5. Dec 27, 2013

### jgens

Unless by space you mean something like vector spaces, it's actually the other way around. The restriction to manifolds is necessary, however, since they are precisely the spaces on which Riemannian metrics are defined.

They are technically different kinds of metrics. The metrics you learn about when studying metric spaces are very different than Riemannian metrics.

6. Dec 27, 2013

### ShayanJ

In fact I was considering the "space" in the OP to mean 3-dimensional Euclidean manifold!
I was starting to feel that way too,because the wikipedia page on Riemannian manifolds were defining Riemannian metrics somehow that I couldn't relate it to the definition of metric in metric spaces!
So I retreat and leave this thread to mathematicians.

7. Dec 27, 2013

### jgens

Riemannian manifolds are those manifolds equipped with a specific Riemannian metric. It can be shown that every manifold can be endowed with such a metric.

Euclidean space has a bit more flexible interpretation in my opinion. Sometimes it can refer to Rn purely as a topological space. Other times it may refer to the vector space structure of Rn. It could mean a combination of the two as well. Or it could refer to Rn as a Riemmanian manifold with the usual metric or something else still.