Derivative of Projection on Manifolds

Question

Let $f \colon X \times Y\to X $ be the projection map, where $X$ and $Y$ are manifolds. I need to show that the derivative on tangent spaces $$Df(x,y) \colon T_x(X) \times T_y(Y)\to T_x(X)$$ will also be the analogous projection.

I am not sure how to start, as I am new to the subject.

As suggested in the comments, this I think is the required isomorphism $D\phi_0×D\psi_0 \to (D\phi_0,D\psi_0)$, where $\phi,\psi$ are local parametrizations of $X$ and $Y$.

Answer 1

Always start with what you know:

1. As $X$ and $Y$ are manifolds, they are equipped with local coordinates $x_i$, $y_i$
2. As a product manifold, $X\times Y$ has a local coordinate system that looks like $(x_i, y_i)$
3. The projection map $f(x,y) = x$

So now, the exercise becomes: write down the differential of the projection in the $(x_i,y_i)$ coordinate system. Can you take it from here?