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Faced with such a task:

Let $k_1, \dots , k_m$ be the normal curvatures of the surface in the directions dividing the plane into angles $\frac{\pi}{m}$ To prove that $k_1 + \dots + k_m = mH/2$, where H - average curvature: $H = \lambda_1 + \lambda_2$

Of course, my first thoughts were that $k_i = \lambda_1 \cos^2 \phi_i \; + \lambda_2 \sin^2 \phi_i$ and instead of $\phi_i \; - \pi/m, 2\pi/m, \dots, (m-1) \pi /m$, substitute these corners and everything would be fine there and I would get what I want to prove. But these are the angles between the velocity vectors, and they lie in the tangent plane, but that formula for $k_i$ is true in the normal plane and moreover in a special basis in which the first quadratic form is singular, and the second is diagonal. Please tell me how to be here and what other ideas can be found to solve this problem?

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In fact, this basis $\{e_1, e_2\}$ lies in tangent space, so everything is good and indeed this solution is correct

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