Curvature of curves through a point of a surface smoothly embedded in Euclidean space

Question

The curve C(𝜃) drawn on a smoothly embedded surface 𝜮 in 3-space — where C(𝜃) is defined as the intersection of 𝜮 with a 2-plane perpendicular to 𝜮 at P — leaving the point P at angle 𝜃 will have the well-known formula for curvature:

k(𝜃) = k₁ cos²(𝜃) + k₂ sin²(𝜃)

at P, where k₁ and k₂ are the principal curvatures of 𝜮 at P.

Likewise, we can look at a point P of a smoothly embedded surface 𝜮 in Euclidean n-space, and consider the curves C(𝜃) that are each the intersection of 𝜮 with a hyperplane that is perpendicular to 𝜮 at P and ask what their curvatures are as a function of angle 𝜃.

Does there exist a formula (like k(𝜃) = k₁ cos²(𝜃) + k₂ sin²(𝜃), or different) for the curvatures k(𝜃) of the curves C(𝜃) in this case?

And what about the special case where 𝜮 is a Riemann surface holomorphically embedded in C^m = R^2m?

Answer 1

Let $u$ be a unit tangent vector at $p$. The curvature $k$ in the direction of $u$ is $|\mathrm{I\!I}_p(u,u)|$, where $\mathrm{I\!I}_p$ is the second fundamental form at $p$; it is a quadratic form on tangent plane $\mathrm{T}_p$ with values in the normal space $\mathrm{N}_p$.

We may write $\mathrm{I\!I}_p$ in a basis $e_1,\dots,e_{n-2}$ of $\mathrm{N}_p$, $$\mathrm{I\!I}_p=s_1\cdot e_1+\dots+s_{n-2}\cdot e_{n-2};$$ so, $$k=\sqrt{s_1(u,u)^2+\dots+s_{n-2}(u,u)^2}.$$

Each form $s_i$ is diagonalizable, denote by $a_i$ and $b_i$ its eigenvalues; let $\phi_i$ be the angle to the first eigenvector. Then we get $$k=\sqrt{\sum_i(a_i\cdot \cos(\theta-\phi_i)^2+b_i\cdot \sin(\theta-\phi_i)^2)^2}.$$ (I do not think you need it, but that is what you asked for.)

By the way, the image of $\mathrm{I\!I}_p$ is at most 3-dimensional subspace of $\mathrm{N}_p$. So you may reduce number of terms to three.