📂Geometry

Differentiable Functions from a Differentiable Manifold to a Differentiable Manifold

Definition¹

Given that $M_{1}, M_{2}$ are each a $n, m$-dimensional differentiable manifold, a mapping $\varphi : M_{1} \to M_{2}$ is defined to be differentiable at $p \in M_{1}$ if it satisfies the following conditions:

1. Whenever a coordinate system $\mathbf{y} : V \subset \mathbb{R}^{m} \to M_{2}$ is given in $\varphi(p)$, there exists a coordinate system $\mathbf{x} : U \subset \mathbb{R}^{n} \to M_{1}$ in $p$ such that $\varphi\left( \mathbf{x}(U) \right) \subset \mathbf{y}(V)$ holds.

2. The mapping $\mathbf{y}^{-1} \circ \varphi \circ \mathbf{x} : U \subset \mathbb{R}^{n} \to \mathbb{R}^{m}$ is differentiable at $\mathbf{x}^{-1}(p)$.

Explanation

Just like when defining differentiable manifolds, differentiation is defined through the coordinate system $\mathbf{x}, \mathbf{y}$.

Condition 1. might look difficult at first, but on a closer look, it precisely matches the definition of the $\epsilon -\delta$ method or the sense of defining continuity in topology.

For condition 2., since $\mathbf{y}^{-1} \circ \varphi \circ \mathbf{x}$ is a function from Euclidean space to Euclidean space, it is differentiable in the classical sense. This mapping is termed the expression of $\varphi$ in coordinate systems $\mathbf{x}$ and $\mathbf{y}$.