I have a source (Recent Developments in Topological Hydrodynamics by Boris Khesin) which talks about the $H^{-1}$ metric on the space $C^\infty(S^1, \text{SO}(3))$. As per my own research, the $H^k$ metric generally refers to $$\left(\sum_{i=0}^k \int \left|f^{(i)}\right|^2\text{d} x\right)^{1/2}.$$
where $f^{(i)}$ is the $i$th derivative of the function $f$, and $\left|\cdot\right|$ is inherited from the metric on the target space (in this case $\text{SO}(3)$).
However, I do not see how this would be generalized to a negative coefficient $k$. I considered interpreting the $-1$th derivative as a primitive function but this has two problems.
- The primitive function will have an arbitrary constant which means I would need some way of picking this constant;
- $S^1$ is not simply connected, so a primitive function may not even exist.
That being said, what metric does $H^{-1}$ refer to?