I'm confused about the relation between two concepts: the push-forward of a smooth map between two smooth manifolds and the differential of a smooth real-valued function.
Let $M,N$ be smooth manifolds and $\varphi :M\to N$ smooth. Then the push-forward of $\varphi$ at $p\in M$ is defined as\begin{align*}\varphi _{*p}:T_pM & \to T_{\varphi (p)}N \\
\varphi _{*p}(X_p)(g) & =X_p(g\circ \varphi )
\end{align*}for $g$ any smooth function on $N$.
On the other hand, for a smooth function $f:M\to \mathbb{R}$, i.e. for the special case $N=\mathbb{R}$, the differential of $f$ at $p$ is defined as\begin{align*}d_pf:T_pM & \to \mathbb{R} \\
d_pf(X_p) & =X_p(f).
\end{align*}Starting from these definitions, can we show that $f_{*p}=d_pf$, or are they different objects?
Best Answer
Welcome to MSE!
I think you are right, actually, in some books the push forward is just called differential even if $N$ is not the real line and the push forward function is denoted by $df$. See even wiki https://en.wikipedia.org/wiki/Pushforward_(differential)
If both $M,N$ are euclidean spaces then push forward $df$ is just Jacobian of a function at that point.
Your case is special for $M$ is a manifold and $N$ is real line, and there differential and push-forward is also the same.
That being said, even if I have called push-forward "differential" for a while, to see the connection in this special case is quite refreshing!