Chern Character of the dual bundle

Question

Let $(E,\nabla)$ be a complex vector bundle with connection. Is there a formula for the Chern character of the dual vector bundle $(E^*,\nabla^*)$ in terms of the chern character/classes of $E$?

Answer 1

If I am correct, there is not a single expression for the Chern character form of the induced connection $\nabla^*$ on the dual bundle $E^*\to M$. But there is still something we can say.

Let $R$ be the curvature of $\nabla$. Write the Chern character form $\textrm{ch}(\nabla)$ into a sum according to its degree, i.e. $$\textrm{ch}(\nabla)=\sum_{k=0}\textrm{ch}_k(\nabla),$$ where $$\textrm{ch}_k(\nabla):=\frac{(-1)^k}{k!(2\pi i)^k}\textrm{tr}(R^k)\in\Omega^{2k}(M).$$ Denote by $R^*$ the curvature of $\nabla^*$. Since $R^*=-R^t$, where $R^t$ means the transpose of $R$, then $$\textrm{ch}_k(\nabla^*)=\frac{(-1)^k}{k!(2\pi i)^k}\textrm{tr}((-R^t)^k)=(-1)^k\frac{(-1)^k}{k!(2\pi i)^k}\textrm{tr}(R^k)=(-1)^k\textrm{ch}_k(\nabla).$$

Thus its "degree components" behave like Chern classes.

Please feel free to let me know if the above calculations are wrong.