📂Geometry

Parallel Vector Fields on Differential Manifolds

Definition¹

Let us consider a differentiable manifold with given affine connection $M$ and $\nabla$. A vector field $V$ following the curve $c : I \to M$ is said to be parallel if it satisfies the following condition.

$$\dfrac{DV}{dt} = 0,\quad \forall i \in I$$

Theorem

Let us consider a differentiable manifold with given affine connection $M$ and $\nabla$. Let $c : T \to M (t\in I)$ be a differentiable curve, and let $V_{0}$ be a tangent vector at $c(t_{0})$.

$$V_{0} \in T_{c(t_{0})}M$$

Then, there exists a unique parallel vector field $V$ following $c$ that satisfies $V(t_{0}) = V_{0}$.

Explanation

The reason why parallel vector fields are useful is because they reduce computation since differentiating them results in $0$.

According to the above theorem, $V(t)$ is represented as below, and thus, the translation of $V(t)$ using $V_{0}$ is referred to as parallel translation.

Proof

Strategy: The proof is accomplished by showing that there is a neighborhood where the parallel vector field exists for all $t \in I$. If we then take the end of that segment as a new starting point, the existence is guaranteed up to some neighborhood from there, which means the theorem holds for the entire region.

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Choose a coordinate system $\mathbf{x} : U \to M$. Assume that $c(I)$ is included in some coordinate neighborhood $\mathbf{x}(U)$.

$$c(t) = \mathbf{x}(c_{1}(t), \dots, c_{n}(t))$$

Let us denote the tangent vector at $t_{0}$ by $V_{0} = \sum_{j} V_{0}^{j} \dfrac{\partial }{\partial x_{j}}$.

• Part 1. Uniqueness

  Now, assume that a parallel vector field $V$ that satisfies $V(t_{0}) = V_{0}$ exists in $\mathbf{x}(U)$. Then, by the definition of parallel vector fields, the following holds:

  $$\begin{align*} 0 = \dfrac{DV}{dt} =&\ \dfrac{D}{dt} \left( \sum_{j} V_{j} \dfrac{\partial }{\partial x_{j}} \right) = \sum_{j} \dfrac{D}{dt} \left( V_{j} \dfrac{\partial }{\partial x_{j}} \right) \\ =&\ \sum_{j} \dfrac{d V_{j}}{d t} \dfrac{\partial }{\partial x_{j}} + \sum_{j} V_{j} \nabla_{\frac{dc}{dt}} \dfrac{\partial }{\partial x_{j}} \end{align*}$$

  At this point, since $\dfrac{dc}{dt} = \sum\limits_{i}\dfrac{d c_{i}}{d t} \dfrac{\partial }{\partial x_{i}}$ and $\nabla_{\frac{\partial }{\partial x_{i}}}\dfrac{\partial }{\partial x_{j}} = \sum_{k} \Gamma_{ij}^{k} \dfrac{\partial }{\partial x_{k}}$, we obtain the following:

  $$\begin{align*} 0 =&\ \sum_{j} \dfrac{d V_{j}}{d t} \dfrac{\partial }{\partial x_{j}} + \sum_{j}V_{j} \nabla_{\sum_{i}\frac{dc_{i}}{dt}\frac{\partial }{\partial x_{i}}} \dfrac{\partial }{\partial x_{j}} \\ =&\ \sum_{j} \dfrac{d V_{j}}{d t} \dfrac{\partial }{\partial x_{j}} + \sum_{i,j}V_{j}\frac{dc_{i}}{dt}\nabla_{\frac{\partial }{\partial x_{i}}} \dfrac{\partial }{\partial x_{j}} \\ =&\ \sum_{j} \dfrac{d V_{j}}{d t} \dfrac{\partial }{\partial x_{j}} + \sum_{i,j}V_{j}\frac{dc_{i}}{dt}\sum_{k} \Gamma_{ij}^{k} \dfrac{\partial }{\partial x_{k}} \\ =&\ \sum_{j} \dfrac{d V_{j}}{d t} \dfrac{\partial }{\partial x_{j}} + \sum_{i,j,k}V_{j}\frac{dc_{i}}{dt} \Gamma_{ij}^{k} \dfrac{\partial }{\partial x_{k}} \\ \end{align*}$$

  Since $j$ is a dummy index, by changing the index of the first term to $k$ and rearranging, we obtain:

  $$0 = \sum_{k} \left( \dfrac{d V_{k}}{d t} + \sum_{i,j}V_{j}\frac{dc_{i}}{dt} \Gamma_{ij}^{k} \right) \dfrac{\partial }{\partial x_{k}}$$

  For this vector to be $0$, all coefficients must be $0$, hence we obtain:

  $$\begin{equation} 0 = \dfrac{d V_{k}}{d t} + \sum_{i,j}V_{j}\frac{dc_{i}}{dt} \Gamma_{ij}^{k}, \quad k=1,\dots,n \end{equation}$$

  This is an ODE system. Therefore, given the initial condition $v_{k}(t_{0}) =V_{0}^{k}$, the uniqueness of $V$ is ensured by Picard’s theorem.

• Part 2. Existence

  Consider an ODE system like $(1)$. Then, by Picard’s theorem, a solution exists for all $t \in I$. Therefore, we can conclude that the described $V$ exists.

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