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Let $(E,g)$ be a metric vector bundle with a connection $\nabla = d +A$, localIy I define a curvature:

$$\tilde R (\nabla)= dA -dA^* - A\wedge A^*- A^* \wedge A$$

where $A^*$ is the transposed of $A$ by the metric $g$.

Then we have $\tilde R(h^* \nabla)= h^{-1} \tilde R(\nabla) h$, with $h\in SO(n)$.

Can we define invariants of manifolds with help of this curvature?

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