Differential of a function that takes values in a submanifold of Euclidean space

Question

When I am trying to compute concretely in differential geometry I sometimes have a function $f\colon M \to \mathbb{R}^n$ let's say, where $M$ is a smooth manifold, but actually the image of $f$ lies in a familiar submanifold $\widetilde{M}$ like the sphere. It usually feels easier to compute the differential in coordinates of $f$ treating $f$ as a function with values in $\mathbb{R}^n$, but when I later want to understand $f$ as a function from $M$ to $\widetilde{M}$, my Jacobian in coordinates is now the wrong size because we treated $f$ as taking values in the $n$-dimensional manifold $\mathbb{R}^n$ instead of the smaller dimensional manifold $\widetilde{M}$. Is there a way to recover the differential of $f$ as a function taking values in $\widetilde{M}$ from this bigger differential?

Answer 1

Let's say we have a map from $M$ to a sphere $S$. Now $S$ has some charts and coordinate systems. There are no clear agreements to what each means, in the sense of : which way do they go: from $S$ to $\mathbb{R}^2$, or the other way. However, call them what you want, you have a map $\chi \colon U \subset \mathbb{R}^2 \to V \subset S^2$, $(\phi, \theta) \mapsto ( \cos \theta \cos \phi, \cos \theta \sin \phi, \sin \theta)$. Now, one sees that $\chi$ maps in fact into $\mathbb{R}^3$. OK.

Now consider a map from $M$ to the sphere. You can think of it as a map from $F\colon M \to U$ ( at least locally). The actual map to the sphere will be $f =\chi \circ F$. Now can you see how the jacobians of $f$ and $F$ are related?