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- TL;DR Summary
- Imagine a 6D flat space. We compactify 3 of them to circles. How does the associated metric tensor look like?

I'm interested in describing a 6-dimensional space of which three are compactified to small circles. Globally this space looks 3-dimensional, like a 2-dimensional cylinder looks 1-dimensional globally.

Kaluza and Klein did a similar thing in the context of 4-dimensional spacetime. They extended the spacetime by attaching a tiny circle to every point. This was expressed in a 5-dimensional metric, containing eight extra off-diagonal metric components. They are the four components of the A four-vector for the electromagnetic field. An extra diagonal component is introduced but that appeared to be unphysical. The extra components imposed a vector bundle on the small circle.

How do we describe a 6D flat space of which we compactify three into small circles? It's easy to describe the metric of a flat 6D space, but how does the metric look like if three of them have been turned to circles? The 2D case would be a 2D flat space of which one dimension is compactified to a circle (a cylinder).

Kaluza and Klein did a similar thing in the context of 4-dimensional spacetime. They extended the spacetime by attaching a tiny circle to every point. This was expressed in a 5-dimensional metric, containing eight extra off-diagonal metric components. They are the four components of the A four-vector for the electromagnetic field. An extra diagonal component is introduced but that appeared to be unphysical. The extra components imposed a vector bundle on the small circle.

How do we describe a 6D flat space of which we compactify three into small circles? It's easy to describe the metric of a flat 6D space, but how does the metric look like if three of them have been turned to circles? The 2D case would be a 2D flat space of which one dimension is compactified to a circle (a cylinder).