- #36
lavinia
Science Advisor
Gold Member
- 3,330
- 715
That is true but if a metric is everywhere locally flat - which is what I was responding to - then the curvature tensor must be identically zero.Dale said:That is true but irrelevant. The curvature is invariant but it need not be the same everywhere in the manifold. The curvature of the exterior Schwarzschild is not zero.
On a manifold with boundary, the boundary manifold inherits a connection from the connection on the entire manifold. This is not the same connection. For instance on a solid ball in Euclidean space with the standard connection, the Riemann curvature tensor is identically zero. The boundary sphere has curvature that it derives from a different connection.
Last edited: