Question

if two families of geodesics surface intersect at a constant angle, prove that the surface has zero gaussian curvature.

Answer 1

Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ₁ and κ₂, at the given point:

$K=k_1k_2$

Principal curvatures at a given point of a surface measure how the surface bends by different amounts in different directions at that point.

So, if geodesics surface intersect at a constant angle, it means that surface has constant bend in one direction. So, one of its principal curvatures = 0.

So, its Gaussian curvature = 0