What really is the modern definition of Euclidean spaces?

Question

What is the modern definition of Euclidean spaces?

I read the Wikipedia article about the topic, but I still don't get it.

Is a Euclidean space

1. something that satisfies the traditional Euclid's axioms, or Hilbert's axioms?

2. or is it defined to be an inner product space?

3. or is it defined to be a set on which we can somehow define the notion of "length" and "angle"?

4. or is it defined to be an affine space?

If a Euclidean space is defined as in case 2 (i.e. as an inner product space), then do we still need Euclid's axioms or Hilbert's axioms?

For example, there's a Hilbert's axiom

For every two points A, B there exists a line a that contains each of the points A, B.

but in terms of inner product space terminology, it can be directly proved by trivial pre-calculus technique, right?

Answer 1

1. This is not widely used nowadays.

2. More precisely it would be a finite-dimensional real inner-product space.

3. This is not specific enough. This is the notion of a Riemannian manifold and includes Euclidean spaces as special cases. The sphere in $\mathbb{R}^3$ is an example of a Riemannian manifold which is not a Euclidean space.

4. A Euclidean space is in particular an affine space and every affine space can be given the structure of a Euclidean space (by choosing an origin and an inner-product). But an affine space does not come with this structure.

Now it depends on the context and in which field of mathematics you are working. Many people view it as in 2 but others have the point of view that Euclidean spaces do not have a distinguished origin. In that case it would be closer to 3 and 4, or more precisely, we would say that it is an affine space with a flat Riemannian metric. Or similarly, a common terminology in differential geometry is:

5. $\mathbb{R}^n$ with the flat metric.