# Local geometry of theory of general relativity

1. Aug 13, 2015

### mikeeey

Hello guys .
Through all the analysis of theory of general relativity we used what so called Manifolds
Manifolds as we know are topological spaces that resemble ( look like) euclidean space locally at tiny portion of space
And an euclidean space is the pair ( real coordinate space R^n , dot product ),
And any euclidean space is flat space,
So manifolds locally are flat , do not have curvature locally
But solutions of GR's equations show the manifolds are not flat locally even they are locally look like euclidean space .
My question is that if the space is curved , Then the Curvature tensor does not depend on the chosen local real coordinate space ( system ) , is it ?!

Thanks .

2. Aug 13, 2015

### Staff: Mentor

Minkowski, not Euclidean - you can find coordinates in which the metric tensor is diagonal with components arbitrarily close to (-1,1,1,1) but not (1,1,1,1) which would be Euclidean.
All tensors are coordinate-independent objects - their value does not depend on the coordinate system. The values of the components of a tensor do change with the coordinate system, but that's just a result of using different coordinate systems to represent the same object - a physics problem may look very different (and be much easier or harder to solve) when it's written in one coordinate system instead of another, but it's still the same problem.

3. Aug 13, 2015

### mikeeey

My question , lets enter to 2-manifolds( surfaces ) if the surface of the shpere(e.g.) is locally look like euclidean space ( made up from gluing small planes together )that mean its curvature locally flat !! , but its not in fact

4. Aug 13, 2015

### Staff: Mentor

That is correct. Somewhere in whatever text you're using you'll find a proper definition of what "locally flat" means. It will be something along the lines of: the difference between the metric tensor and the flat-space metric tensor can be made arbitrarily small by considering a small enough region.

5. Aug 13, 2015

### mikeeey

Last question , how does the curvature tensor of the surface of the sphere is not zero and still locally flat ?!

Thanks

6. Aug 13, 2015

### Staff: Mentor

The curvature tensor is not zero, but a zero curvature tensor is not a requirement for local flatness. Local flatness means that a sufficiently small region can be approximated as flat, and the smaller you make the region the better the approximation is. Whatever text you're using should have a proper formal definition - keep looking until you find it and understand it.

7. Aug 13, 2015

Thank you