In Definition 2.5 of Riemannian Geometry by do Carmo, the following definition is given:
Let $M_1^n$ and $M_2^m$ be differentiable manifolds. A mapping $\varphi: M_1 \to M_2$ is differentiable at $p \in M_1$ if given a parameterization $\mathbf{y}: V \subset \mathbb{R}^m \to M_2$ at $\varphi(p)$ there exists a parameterization $\mathbf{x}: U \subset \mathbb{R}^n \to M_1$ at $p$ such that $\varphi(\mathbf{x}(U)) \subset \mathbf{y}(V)$ and the mapping $$\mathbf{y}^{-1} \circ \varphi \circ \mathbf{x}: U \subset \mathbb{R}^n \to \mathbb{R}^m$$ is differetiable at $\mathbf{x}^{-1}(p)$.
Afterwards, do Carmo states that the definition is independent of the parameterizations chosen, but I'm not sure what he means by that. My understanding is that the definition requires that such a parameterization $\mathbf{x}$ exist for every such parameterization $\mathbf{y}$. Which part of the definition is supposed to be independent of the $\mathbf{x}$ and $\mathbf{y}$ chosen?
Best Answer
Consider any parametrizations $y_i : V_i \subseteq \mathbb{R}^m \to M_2$, $x_i : U_i \subseteq \mathbb{R}^n \to M_1$ with $p \in x_i(U_i)$, $\phi(x_i(U_i)) \subseteq y_i(V_i)$ for $i = 1, 2$. Let $diff_i$ abbreviate the statement that $y_i^{-1} \circ \phi \circ x_i$ is differentiable at $x_i^{-1}(p)$.
Do Carmo’s claim is that $diff_1 \iff diff_2$. This is straightforward to prove from the definition of $diff_i$, the change—of-parameters property, and the fact that the composition of differentiable functions is differentiable (when we use our prior definition of differentiability for subsets of Euclidean space).
Note that continuity of $\phi$ at $p$ is equivalent to saying that for all parametrizations $y : V \to M_2$, with $\phi(p) \in y(V)$, there exists a parametrisation $x : U \to M_1$ with $p \in x(U) \subseteq \phi^{-1}(y(V))$. So you cannot take the existence of some such $x$ for granted. I don’t have my copy of do Carmo with me, so I’m not sure whether he’s discussed continuity yet, but this is the natural definition corresponding to the “obvious” choice of a topology for a manifold - the one where basic opens are of the form $x(U)$ for a parametrisation $x : U \subseteq \mathbb{R}^n \to M_1$.