📂Geometry

Necessary and Sufficient Conditions for a Vector Field to be Parallel Along a Curve

Theorem¹

Let $\boldsymbol{\gamma}(t) = \mathbf{x}\left( \gamma^{1}(t), \gamma^{2}(t) \right)$ be a regular curve on $\mathbf{x}$, the coordinate patch. Let $\mathbf{X}(t)$ be a differentiable vector field along the curve $\boldsymbol{\gamma}$.

$$\mathbf{X} = X^{1}\mathbf{x}_{1} + X^{2}\mathbf{x}_{2}$$

Then, the necessary and sufficient condition for $\mathbf{X}(t)$ to be parallel along $\boldsymbol{\gamma}$ is as follows.

$$0 = \dfrac{d X^{k}}{d t} + \sum_{i,j} \Gamma_{ij}^{k} X^{i}\dfrac{d \gamma^{j}}{d t},\quad k=1,2$$

Proof

The definition of $\mathbf{X}$ being parallel is that $\mathbf{X}_{t}$ is orthogonal to the surface. Therefore, since $\mathbf{x}_{l}$ is the basis of the tangent plane, the following holds.

$$\mathbf{X} \text{ is parallel } \iff 0 = \left\langle \dfrac{d \mathbf{X}}{d t}, \mathbf{x}_{l} \right\rangle$$

When we calculate $\mathbf{X}_{t}$, it is as follows.

$$\dfrac{d \mathbf{X}}{d t} = \dfrac{d}{dt}\left( \sum_{i}X^{i}\mathbf{x}_{i}\right) = \sum_{i} \dfrac{d X^{i}}{d t} \mathbf{x}_{i} + \sum_{i} X^{i}\dfrac{d \mathbf{x}_{i}}{d t} = \sum_{i} \dfrac{d X^{i}}{d t} \mathbf{x}_{i} + \sum_{i,j} X^{i}\mathbf{x}_{ij}\dfrac{d \gamma^{j}}{d t}$$

Therefore

$$\begin{align} \mathbf{X} \text{ is parallel } \iff 0 =&\ \left\langle \sum_{i} \dfrac{d X^{i}}{d t} \mathbf{x}_{i} , \mathbf{x}_{l} \right\rangle + \left\langle \sum_{i,j} X^{i}\mathbf{x}_{ij}\dfrac{d \gamma^{j}}{d t} , \mathbf{x}_{l} \right\rangle \nonumber \\ =&\ \sum_{i} \dfrac{d X^{i}}{d t} \left\langle \mathbf{x}_{i} , \mathbf{x}_{l} \right\rangle + \sum_{i,j}\left\langle \mathbf{x}_{ij} , \mathbf{x}_{l} \right\rangle X^{i}\dfrac{d \gamma^{j}}{d t} \nonumber \\ =&\ \sum_{i} \dfrac{d X^{i}}{d t} g_{il} + \sum_{i,j}\left\langle \mathbf{x}_{ij} , \mathbf{x}_{l} \right\rangle X^{i}\dfrac{d \gamma^{j}}{d t} \label{dfsdf} \end{align}$$

Here, $g_{il}$ is the coefficient of the first fundamental form. Now, multiplying both sides of the above equation by $g^{lk}$ and summing over the index $l$ gives the following.

$$0 = \sum_{i,l} \dfrac{d X^{i}}{d t} g_{il}g^{lk} + \sum_{i,j,l}\left\langle \mathbf{x}_{ij} , \mathbf{x}_{l} \right\rangle g^{lk} X^{i}\dfrac{d \gamma^{j}}{d t}$$

Then, by the properties of $g_{il}g^{lk}$ and the definition of the Christoffel symbols, the following holds.

$$\begin{align*} 0 =&\ \sum_{i} \dfrac{d X^{i}}{d t} \delta_{i}^{k} + \sum_{i,j} \Gamma_{ij}^{k} X^{i}\dfrac{d \gamma^{j}}{d t} \\ =&\ \dfrac{d X^{k}}{d t} + \sum_{i,j} \Gamma_{ij}^{k} X^{i}\dfrac{d \gamma^{j}}{d t},\quad k=1,2 \end{align*}$$

Conversely, multiplying both sides of the above equation by $g_{kl}$ and summing over the index $k$ gives $(1)$.

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