📂Geometry

Gauss Curvature and Geodesic Curvature

Buildup¹

$$\left\{ T(s), N(s), B(s), \kappa (s), \tau (s) \right\}$$

Recall how we used the Frenet-Serret apparatus when analyzing curves. When studying surfaces, we will consider similar concepts. When $\boldsymbol{\alpha}$ is the unit speed curve, the curvature of the curve was defined as the magnitude of acceleration $\kappa = \left| T^{\prime} \right| = \left| \boldsymbol{\alpha}^{\prime \prime} \right|$. It is natural to think about how curved a surface is by looking at how curved the curves on the surface are.

Consider the surface given as $\mathbf{x} : U\subset \R^{2} \to M$. Let $\boldsymbol{\alpha}(s)$ be the unit speed curve on the simple surface $\mathbf{x}$. Then let’s denote the Frenet-Serret apparatus for $\boldsymbol{\alpha}$ as follows.

$$\left\{ \mathbf{T}, \mathbf{N}, \mathbf{B}, \kappa, \tau \right\}$$

Let’s call the unit normal at point $p \in M$ as $\mathbf{n}$. Let’s define the set of all vectors that are point $p$ as $N_{p}M$.

$$N_{p}M := \left\{ r \mathbf{n} : r \in \R \right\} = \left\{ \text{all vectors perpendicular to } M \text{ at } p \right\}$$

Then by definition of the tangent plane, $T_{p}M$ is the orthogonal complement of $N_{p}M$.

$$N_{p}M ^{\perp} = T_{p}M$$

Therefore, $\R^{3}$ is orthogonally decomposed as follows, and $\boldsymbol{\alpha}^{\prime \prime}$ can be expressed as the linear combination of vectors of the two spaces.

$$\R^{3} = N_{p}M \oplus T_{p}M \quad \text{and} \quad \boldsymbol{\alpha}^{\prime \prime}(s) = n_{1}\mathbf{n}(s) + n_{2}\mathbf{n}^{\perp}(s)\quad (\mathbf{n}\in N_{p}M,\ \mathbf{n}^{\perp}\in T_{p}M)$$

Let’s call $\mathbf{T} = \boldsymbol{\alpha}^{\prime}$ the tangent vector. Since $\boldsymbol{\alpha}$ is the unit speed vector, the following equation holds.

$$\left| \boldsymbol{\alpha}^{\prime}(s) \right|^{2} = \left| \mathbf{T}(s) \right|^{2} = \left\langle \mathbf{T}, \mathbf{T} \right\rangle = 1$$

Differentiating both sides gives the following result by the derivative of dot product.

$$\begin{align*} && \left\langle \mathbf{T}, \mathbf{T} \right\rangle^{\prime} =&\ 0 \\ \implies && \left\langle \mathbf{T}^{\prime}, \mathbf{T} \right\rangle =&\ 0 \\ \implies && \left\langle \boldsymbol{\alpha}^{\prime \prime}, \mathbf{T} \right\rangle =&\ 0 \end{align*}$$

Therefore, $\boldsymbol{\alpha}^{\prime \prime}$ is perpendicular to $\mathbf{T}$. Separating $\boldsymbol{\alpha}^{\prime \prime}$, since $\mathbf{n}$ and $\mathbf{T}$ are perpendicular to each other, we obtain the following.

$$\begin{align*} && \left\langle \boldsymbol{\alpha}^{\prime \prime}, \mathbf{T} \right\rangle =&\ 0 \\ \implies && \left\langle n_{1}\mathbf{n} + n_{2}\mathbf{n}^{\perp}, \mathbf{T} \right\rangle =&\ 0 \\ \implies && \left\langle n_{1}\mathbf{n}, \mathbf{T} \right\rangle + \left\langle n_{2}\mathbf{n}^{\perp}, \mathbf{T} \right\rangle =&\ 0 \\ \implies && \left\langle n_{2}\mathbf{n}^{\perp}, \mathbf{T} \right\rangle =&\ 0 \end{align*}$$

Therefore, it can be known that $\mathbf{n}^{\perp}$ is a vector perpendicular to both $\mathbf{n}$ and $\mathbf{T}$. Let’s define the vector $\mathbf{S}$ as follows.

$$\mathbf{S} := \mathbf{n}\times \mathbf{T} \quad \text{and} \quad \boldsymbol{\alpha}^{\prime \prime} = n_{1}\mathbf{n} + s\mathbf{S}$$

$\mathbf{S}$ is called the intrinsic normal of $\boldsymbol{\alpha}$.

Definition

The component $n_{1}$ of $\mathbf{n}$ is called the normal curvature of the unit speed curve $\boldsymbol{\alpha}$, denoted as $\kappa_{n}$.

$$\kappa_{n} := \left\langle \boldsymbol{\alpha}^{\prime \prime}, \mathbf{n} \right\rangle$$

The component $s$ of $\mathbf{S}$ is called the geodesic curvature of the unit speed curve $\boldsymbol{\alpha}$, denoted as $\kappa_{g}$.

$$\kappa_{g} := \left\langle \boldsymbol{\alpha}^{\prime \prime}, \mathbf{S} \right\rangle$$

Therefore, the following equation holds.

$$\kappa (s) \mathbf{N}(s) = \mathbf{T}^{\prime}(s) = \boldsymbol{\alpha}^{\prime \prime}(s) = \kappa_{n}(s)\mathbf{n}(s) + \kappa_{g}(s)\mathbf{S}(s)$$

Explanation

The normal curvature $\kappa_{n}$ is used to measure how much the surface $M$ is curved at $\R^{3}$. The geodesic curvature $\kappa_{g}$ is used to measure how much the curve $\boldsymbol{\alpha}$ is curved on the surface $M$. For example, a curve with the geodesic curvature $\kappa_{g}$ equal to $0$ implies a straight line on the surface, i.e., a geodesic.

Since $\mathbf{n}, \mathbf{S}$ is a unit vector, the following equation holds according to the definition above.

$$\kappa^{2} = \kappa_{n}^{2} + \kappa_{g}^{2}$$