Gaussian curvature is usually defined as the product of maximal and minimal normal curvatures of curves through a point on a surface. What if we use geodesic curvatures of the ambient space instead?
For example, the geodesics of a horosphere are horocycles in the ambient hyperbolic space, which have geodesic curvature $1$. Therefore, the product of the maximal and minimal such curvatures are $1$ for any point on a horosphere. This is different from the Gaussian curvature, which is $0$. Interestingly, $-1$ + $1$ = $0$, suggesting that the sectional curvature of the ambient space plus the curvature I defined equals the Gaussian curvature.
Does this type of curvature have a name, or coincide with some other concept of curvature?