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- Can we use the Riemann curvature tensor to identify different manifolds?

Hi. I have a question.

Two manifolds can be equivalent but have different co-ordinates and a correspondingly different metric defined on them.

What do I mean by "equivalent"? The manifolds are diffeomorphic so that the metrics are just the usual tensor transformations of each other due to a change in co-ordinates and the manifolds themselves are smoothly and bijectively mapped to/from each other by the co-ordinate transformation. However, I'll take whatever type of identification and sense of equivalence may be possible.

It doesn't seem to be as simple as just using the Riemann curvature tensor to identify different spaces but this is the kind of thing I was hoping for.

Here's an example:

The Riemann curvature vanishes

<--> There exists co-ordinates such that the metric has constant components.

This is great and it's almost enough to identify and categorise the space but not quite enough (for example the metric could have any signature).

Two manifolds can be equivalent but have different co-ordinates and a correspondingly different metric defined on them.

**Is there an easy way to identify when two such manifolds are equivalent but just have different co-ordinates?**What do I mean by "equivalent"? The manifolds are diffeomorphic so that the metrics are just the usual tensor transformations of each other due to a change in co-ordinates and the manifolds themselves are smoothly and bijectively mapped to/from each other by the co-ordinate transformation. However, I'll take whatever type of identification and sense of equivalence may be possible.

It doesn't seem to be as simple as just using the Riemann curvature tensor to identify different spaces but this is the kind of thing I was hoping for.

Here's an example:

The Riemann curvature vanishes

<--> There exists co-ordinates such that the metric has constant components.

This is great and it's almost enough to identify and categorise the space but not quite enough (for example the metric could have any signature).