Let $E\to X$ be a complex flat vector bundle, and say $\nabla_0$ and $\nabla_1$ are two flat connections on it. Let $p:X\times[0, 1]\to X$ denote the projection onto the first factor. Is there a way to construction a flat connection $\tilde{\nabla}$ on $p^*E\to X\times[0, 1]$ such that $\tilde{\nabla}|_{X\times 0}=\nabla_0$ and $\tilde{\nabla}|_{X\times 1}=\nabla_1$?

If this is impossible, then is it possible to construction a (not necessarily flat) connection $\nabla'$ on $p^*E\to X\times[0, 1]$ such that $\tilde{\nabla}|_{X\times 0}=\nabla_0$, $\tilde{\nabla}|_{X\times 1}=\nabla_1$ and $\textrm{ch}(\nabla')=$ the rank of $p^*E\to X\times[0, 1]$? Here $\textrm{ch}(\nabla')$ is the Chern character form of $\nabla'$.

Thanks.

Let $E\to X$ be a complex flat vector bundle, and say $\nabla_0$ and $\nabla_1$ are two flat connections on it. Let $p:X\times[0, 1]\to X$ denote the projection onto the first factor. Is there a way to construct a flat connection $\tilde{\nabla}$ on $p^*E\to X\times[0, 1]$ such that $\tilde{\nabla}|_{X\times 0}=\nabla_0$ and $\tilde{\nabla}|_{X\times 1}=\nabla_1$?

If this is impossible, then is it possible to construct a (not necessarily flat) connection $\nabla'$ on $p^*E\to X\times[0, 1]$ such that $\tilde{\nabla}|_{X\times 0}=\nabla_0$, $\tilde{\nabla}|_{X\times 1}=\nabla_1$ and $\textrm{ch}(\nabla')=$ the rank of $p^*E\to X\times[0, 1]$? Here $\textrm{ch}(\nabla')$ is the Chern character form of $\nabla'$.

Thanks.