I am working on a problem in geometric analysis and I arrived at the following question: what would be the minimum of regularity I could require from a metric in order to define a first fundamental form on an abstract surface (two-dimensional manifold) with Gaussian curvature ${\cal K}=-1$?
More precisely, let us consider the following: assume the existence of $C^1$ functions $f_{ij}=f_{ij}(x,t),\,\,1\leq i\leq 3,\,\,1\leq j\leq 2$, defined on an open set $U\subseteq{\mathbb R}^2$ (non-empty, of course) such that the one-forms $\omega_i=f_{i1}dx+f_{i2}dt,\quad 1\leq i\leq 3$, satisfy the equations \begin{equation} d\omega_1=\omega_3\wedge\omega_2,\quad d\omega_2=\omega_1\wedge\omega_3,\quad d\omega_3=\omega_1\wedge\omega_2.\quad\quad (1) \end{equation}
The equations above make sense with the required regularity and, as a result, $I=\omega_1^2+\omega_2^2$ would define a $C^1$ first fundamental form ($C^1$-metric) for a manifold with Gaussian curvature ${\cal K}=-1$.
The problem is that, usually, in differential geometry one requires the metric as being a tensor with $C^2$ components, e.g, see Theorem 4.24, page 153, in [W. Kühnel, Differential Geometry, 3nd Ed., AMS, (2015)]. Actually, most of the books I studied consider $C^\infty$ metrics. This one by Künhel is one of the few I found requiring finite regularity, but being $C^2$ would contradict what I wrote above. On the other hand, on page 760 of the paper [P. Hartman and A. Wintner, On the fundamental equations of differential geometry, Amer. J. Math., vol. 72, 757--774, (1950)], the authors consider $C^1$ metrics, but in conjunction with a $C^0$ second fundamental form.
My questions is: Would it be possible to require $C^1$ regularity for the metric (or the forms satisfying (1)) and still have a well defined Gaussian curvature? The paper by Hartman and Wintner apparently supports a positive answer, but the fact that usually books on differential geometry require higher regularity makes me worry about if I could or not have it.