Vector Field on Differentiable Manifold 📂Geometry

Vector Field on Differentiable Manifold

Buildup¹

Consider the easy definition of a vector field. In 3-dimensional space, a vector field is a function $X : \mathbb{R}^{3} \to \mathbb{R}^{3}$ that maps a 3-dimensional vector to another 3-dimensional vector. When considering this in the context of manifolds, $X$ maps a point $\mathbb{R}^{3}$ on the differential manifold $p$ to a vector $\mathbb{R}^{3}$ in $\mathbf{v}$, treating this vector $\mathbf{v}$ as an operator to consider as a directional derivative (= tangent vector). Therefore, a vector field is a function that maps a point $\mathbb{R}^{3}$ on a manifold $p$ to a tangent vector $p$ at $\mathbf{v}_{p} \in T_{p}\mathbb{R}^{3}$.

The codomain of a vector field is then the set of all tangent vectors at every point. Thus, a vector field $X$ is defined as the following function.

$$ X : \mathbb{R}^{3} \to \bigcup \limits_{p\in \mathbb{R}^{3}} T_{p}\mathbb{R}^{3} $$

To generalize this concept to manifolds, let’s define the tangent bundle $M$ of a differential manifold $TM$ as follows.

$$ TM := \bigsqcup \limits_{p\in M} T_{p}M $$

Here, $\bigsqcup$ is a disjoint union.

Definition

A vector field $M$ on a differential manifold $X$ is a function that maps each point $p \in M$ to a tangent vector $p$ at $X_{p} \in T_{p}M$.

$$ \begin{align*} X : M &\to TM \\ p &\mapsto X_{p} \end{align*} $$

Explanation

Values of a Vector Field

Considering the definition of the tangent bundle, the element of $TM$ is $(p, X_{p})$, but it is mentioned in the definition that it maps $X_{p}$, which can raise questions.

$$ \begin{equation} TM := \bigsqcup \limits_{p \in M } T_{p}M = \bigcup_{p \in M} \left\{ p \right\} \times T_{p}M = \left\{ (p, X_{p}) : p \in M, X_{p} \in T_{p}M \right\} \end{equation} $$

So, to be precise, according to the definition of the disjoint union, an element of $TM$ is indeed the ordered pair $(p, X_{p})$, but it is essentially treated as if it were $X_{p}$.

Thinking again about the definition of the tangent bundle, what we really want to do is not just collect ordered pairs $(p, X_{p})$ but to collect all tangent vectors at each point $p$. However, since each of $T_{p}M$ is isomorphic to $\mathbb{R}^{n}$, there can be ambiguity when doing the union.

$$ T_{p}M \approxeq \mathbb{R}^{n} \approxeq T_{q}M $$

For example, if $M$ is a 3-dimensional manifold, there is ambiguity in treating the vector $T_{p}M \approxeq \mathbb{R}^{3}$ represented from $\begin{bmatrix} 1 & 1 & 1\end{bmatrix}^{T}$ and the vector $X_{p}$ represented from $T_{q}M \approxeq \mathbb{R}^{3}$ as the same. Therefore, defining $\begin{bmatrix} 1 & 1 & 1\end{bmatrix}^{T}$ as a set of ordered pairs is to make clear that $X_{q}$ and $TM$ are not the same and are distinctly different. From this, it naturally leads to considering a bijective function like $X_{p}$, treating it as $X_{q}$.

In some textbooks, to avoid this detailed explanation or assuming that readers adequately understand, the tangent bundle $\iota_{p} : (p, X_{p}) \mapsto X_{p}$ is sometimes defined as follows.

$$ TM := \bigcup\limits_{p\in M} T_{p}M = \left\{ X_{p} \in T_{p}M : \forall p \in M \right\} $$

Of course, as reiterated, the above definition and $(p, X_{p}) \approx X_{p}$ are essentially the same. Also, note that the function value of $TM$ according to the above definition is a function $(1)$.

$$ X_{p} : \mathcal{D} \to \mathbb{R} $$

Vector Field as an Operator

Let’s say $X$ is an $X_{p}$-dimensional differential manifold. Let the set of differentiable functions on $M$ be called $n$.

$$ \mathcal{D} = \mathcal{D}(M) := \left\{ \text{all real-valued functions of class } C^{\infty} \text{ defined on } M \right\} $$