📂Geometry

Properties of the Second Normal Form

Definition

The second fundamental form is defined as a bilinear form on the tangent space $T_{p}M$. For two tangent vectors $\mathbf{X}=\sum X^{i}\mathbf{x}_{i}$ and $\mathbf{Y} = \sum Y^{j}\mathbf{x}_{j}$, it is defined as:

$$II ( \mathbf{X}, \mathbf{Y}) = \sum _{i,j} L_{ij} X^{i} Y^{j}$$

where the coefficients $L_{ij}$ are as follows.

$$L_{ij} = \left\langle \mathbf{x}_{ij}, \mathbf{n} \right\rangle$$

Properties¹

1. $II$ is symmetric.

2. If $\mathbf{T}$ is the tangent field of a unit speed curve $\boldsymbol{\gamma}$, then $\kappa_{n} = II (\mathbf{T}, \mathbf{T})$ holds, where $\kappa_{n}$ is the normal curvature.

3. Let’s say $\boldsymbol{\alpha}, \boldsymbol{\beta}$ is a regular curve for which $\boldsymbol{\alpha}(0) = \boldsymbol{\beta}(0)$ holds. If the velocity vectors of the two curves satisfy $\boldsymbol{\alpha}^{\prime}(0) = \lambda \boldsymbol{\beta}^{\prime}(0)$ for $\lambda \ne 0$, then the normal curvature $\kappa_{n}$ of the two curves is the same when $t=0$.

Explanation

3. While it’s described for the case where $t=0$, it naturally generalizes to any $t$.

   Since the tangent is the magnitude of the velocity vector made to $1$, the fact that the velocity vector is a constant multiple means $T_{\boldsymbol{\alpha}} = \pm T_{\boldsymbol{\beta}}$ holds for the tangent of the two curves.

   This implies that curves with tangents in the same direction have the same normal curvature, indicating that the normal curvature is determined solely by the tangent, not dependent on the curve itself.

Proof

Property 1

That $II$ is symmetric means either $II( \mathbf{X}, \mathbf{Y} ) = II( \mathbf{Y}, \mathbf{X} )$ or $L_{ij} = L_{ji}$ holds. Assuming the coordinate chart mapping $\mathbf{x}$ is sufficiently smooth, then $\mathbf{x}_{ij} = \mathbf{x}_{ji}$ holds. Therefore,

$$L_{ij} = \left\langle \mathbf{x}_{ij}, \mathbf{n} \right\rangle = \left\langle \mathbf{x}_{ji}, \mathbf{n} \right\rangle = L_{ji}$$

■

Property 2

Let’s consider $\boldsymbol{\gamma}(s) = \mathbf{x}\left( \gamma^{1}(s), \gamma^{2}(s) \right)$ to be a unit speed curve. Then the tangent, by the chain rule, is as follows.

$$\begin{align*} \mathbf{T} &= \dfrac{d \boldsymbol{\gamma}}{d s} = \dfrac{d }{d s}\mathbf{x}\left( \gamma^{1}(s), \gamma^{2}(s) \right) \\ &= \dfrac{\partial \mathbf{x}}{\partial \gamma^{1}}\dfrac{\partial \gamma^{1}}{\partial s} + \dfrac{\partial \mathbf{x}}{\partial \gamma^{2}}\dfrac{\partial \gamma^{2}}{\partial s} \\ &= (\gamma^{1})^{\prime}\mathbf{x}_{1} + (\gamma^{2})^{\prime}\mathbf{x}_{2} \\ &= T^{1}\mathbf{x}_{1} + T^{2}\mathbf{x}_{2} \end{align*}$$

$$\implies T^{i} = (\gamma^{i})^{\prime}$$

Auxiliary Lemma

For any unit speed curve $\boldsymbol{\gamma}(s) = \mathbf{x}\left( \gamma^{1}(s), \gamma^{2}(s) \right)$,

$$\kappa_{n} = \sum \limits_{i=1}^{2} \sum \limits_{j=1}^{2} L_{ij} (\gamma^{i})^{\prime} (\gamma^{j})^{\prime}$$

Thus, by the definition of the second fundamental form and the above auxiliary lemma, the following holds:

$$II(\mathbf{T}, \mathbf{T}) = \sum\limits_{i, j} L_{ij}T^{i}T^{j} = \sum \limits_{i, j} L_{ij} (\gamma^{i})^{\prime} (\gamma^{j})^{\prime} = \kappa_{n}$$

■

Property 3

Let’s denote the tangent vectors at $t=0$ of $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ as $T_{\boldsymbol{\alpha}}, T_{\boldsymbol{\beta}}$, respectively. Since it’s assumed that $\boldsymbol{\alpha} ^{\prime}(0) = \lambda \boldsymbol{\beta} ^{\prime}(0)$, the following holds:

$$T_{\boldsymbol{\alpha}} = \pm T_{\boldsymbol{\beta}}$$

Therefore, the following is true:

$$II ( T_{\boldsymbol{\alpha}}, T_{\boldsymbol{\alpha}}) = II ( \pm T_{\boldsymbol{\beta}}, \pm T_{\boldsymbol{\beta}}) = II ( T_{\boldsymbol{\beta}}, T_{\boldsymbol{\beta}})$$

Hence, by property 2, the normal curvature of the two curves when $t=0$ is the same.

■