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