Let (E,g) be a metric vector bundle with a connection ∇=d+A, localIy I define a curvature:
˜R(∇)=dA−dA∗−A∧A∗−A∗∧A
where A∗ is the transposed of A by the metric g.
Then we have ˜R(h∗∇)=h−1˜R(∇)h, with h∈SO(n).
Can we define invariants of manifolds with help of this curvature?