Local flat space and the Riemann tensor

[nigelscott]

Can somebody explain in layman's terms the connection between local flat space (tangent planes) on a manifold and the Riemann tensor.

 

[bcrowell]

I don't think laymen know the terms you're using. What is your background? E.g., are you a university sophomore majoring in physics, ...?

 

[nigelscott]

No, just a person with an interest in this area. I studied solid state physics (QM) in college many years ago and like to keep my mind active by thinking about this stuff.

My understanding is that if I pick a point on a manifold that in the limit can be considered as being flat, I can use cartesian coordinates. If I uses cartesion coordinates the derivative of the metric tensor is 0 and covariant derivative become the ordinary derivative. So doesn't this imply that the Riemann tensor is not defined at that point? What am I not understanding?

 

[bcrowell]

If you want to visualize the tangent plane, you can visualize it as what it says: a plane that is tangent to the manifold. That means you can't visualize it as a small region of the manifold itself. The tangent plane isn't curved, it's flat, so you can't measure curvature within it.

My understanding is that if I pick a point on a manifold that in the limit can be considered as being flat, I can use cartesian coordinates.

To talk about curvature, you need more than a point, you need a neighborhood around the point. This is the same as in freshman calculus. If I tell you that the function f passes through the origin, that doesn't help you to calculate its curvature (second derivative) at the origin.

The connection between the tangent plane and the Riemann tensor would be that the Riemann tensor is an operator that works with vectors taken from the tangent plane. If you take a vector from the tangent plane and transport it around a little parallelogram, it's changed by the time it comes back to the start. There are four vectors from the tangent plane involved here: (1) the original vector, (2) the change in the original vector, (3) and (4) two vectors that describe the parallelogram. This is why the Riemann tensor has four indices.

 

[nigelscott]

OK, getting closer. So I can work I can work in Euclidean space on the tangent plane but if I want to get an idea of the curvature of space in that neighborhood I have to parallel transport one of the vectors from the tangent plane around a small loop and measure the angle change. Is this typically how problems involving manifolds are solved? i.e. work out things in simpler flat space time and then figure out how curvature impacts things.