Consider the trivial fibration $\mathbb{T}^2\to\mathbb{S}^1$, where $\mathbb{T}^2$ is the two-torus. Denote by $\mathbb{C}\to\mathbb{T}^2$ the trivial line bundle over $\mathbb{T}^2$, and equip it with the trivial connection. The cohomology bundle $H(\mathbb{S}^1, \mathbb{C}|_{\mathbb{S}^1})\to\mathbb{S}^1$ is then a trivial bundle (well every bundle over $\mathbb{S}^1$ is trivial). There is an induced connection, the Gauss-Manin connection $\nabla^{H(\mathbb{S}^1, \mathbb{C}|_{\mathbb{S}^1})}$, on $H(\mathbb{S}^1, \mathbb{C}|_{\mathbb{S}^1})\to\mathbb{S}^1$.
Does it make sense to ask the even (not odd) Chern character form $\textrm{ch}(\nabla^{H(\mathbb{S}^1, \mathbb{C}|_{\mathbb{S}^1})})$? Should $\textrm{ch}(\nabla^{H(\mathbb{S}^1, \mathbb{C}|_{\mathbb{S}^1})})$ be 2, the rank of $H(\mathbb{S}^1, \mathbb{C}|_{\mathbb{S}^1})\to\mathbb{S}^1$?
