Sets of Differentiable Real-Valued Functions on a Differentiable Manifold 📂Geometry

Sets of Differentiable Real-Valued Functions on a Differentiable Manifold

Definition¹

Let $M$ be a differentiable manifold. The set of differentiable functions $f : M \to \mathbb{R}$ at point $p \in M$ is denoted as $\mathcal{D}$.

$$ \mathcal{D} := \left\{ \text{all real-valued functions on } M \text{ that are differentialable at } p \right\} $$

On $M$, the set of differentiable functions $f : M \to \mathbb{R}$ is denoted as $\mathcal{D}(M)$.

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

If the sum and product in $\mathcal{D}(M)$ are defined pointwise, then $\mathcal{D}(M)$ becomes a ring.

$$ \begin{align*} (f + g)(p) &= f(p) + g(p) \\ (fg)(p) &= f(p)g(p) \end{align*} \qquad \forall f, g \in \mathcal{D}(M) $$

Since the codomain of $f, g$ is $\mathbb{R}$, $f(p) + g(p)$, and $f(p)g(p)$ are well-defined as the sum and product of real numbers.