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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?

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  • $\begingroup$ For $\mathbb R^2$ surfaces geodesic polar coordinates can be employed to describe constant $k_g$ ovals on constant $K$ surfaces. $K, k_g$ can be expressed in terms of Christoffel symbols from first fundamental form. Also we have max, min normal curvatures $k_n$ for zero geodesic torsion. $\endgroup$ – Narasimham Feb 26 at 8:48
  • $\begingroup$ Normal curvature depends only on the direction of the tangent vector, not on the actual curve. This is far from true for geodesic curvature. What do you mean by "geodesic curvatures of the ambient space"? This makes no sense to me. You can choose curves with geodesic curvature arbitrarily large (or small). $\endgroup$ – Ted Shifrin Feb 27 at 1:29

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