Tensor product of flat connections is flat
Suppose is a differential manifold and are vector bundles over . Suppose are connections on respectively. Suppose, further, that the Riemann curvature tensor of as well as of is zero. In other words, both and are flat connections. Then, the tensor product of connections is also a flat connection.
- Formula for curvature of tensor product of connections: This states that:
Proof using the formula for curvature of tensor product
By the formula for tensor product of connections, we have:
The right side is zero. Thus, is identically zero on all pure tensors. Further, since is a tensor, at a point depends only on the value of at that point. Thus, at every point, we have shown that is identically zero on all pure tensors. Since every tensor is a sum of pure tensors and is linear, is identically zero at each point, and hence identically zero as a tensor. Thus, is flat.