Is the following Law of Sines valid on all surfaces isometric to a sphere?
$$\frac{\sin A }{ \sin a }= \cdots = \frac{ \sin C }{ \sin c } = E.$$
And similarly,
Is the following Law of hyperbolic Sines valid on all surfaces isometric to a pseudo-sphere?
$$\frac{\sin A }{ \sinh a }= \cdots = \frac{ \sin C }{ \sinh c} = H.$$
If so, what is this non-dimensional quantity ( $E$ or $H$ ) remaining invariant in isometric mappings that we can geometrically identify ?
The basis of this question is conservation of geodesic curvature (one among first fundamental form dependents) on constant Gauss curvature surfaces (here zero for a geodesic).
Narasimham GL