Push forward or differential : is there a link with the differential of a function?
They're the same idea. If $\varphi:M\rightarrow N$ is a function between smooth manifolds, the differential $d\varphi(x)$ is, at each point, the best linear approximation to $\varphi$.
- In real space, we have that for a smooth function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$, the differential $df(r, \Delta r)$ is a linear approximation of $f$ such that if you pick a point $r$ and (small) change $\Delta r$ in $\mathbb{R}^m$, you have that the differential is (1) a linear function which (2) approximates the change in the value of the function as you deviate slightly from $r$:
$$ f(r+\Delta r) - f(\Delta r) \approx f^\prime(r)\Delta r = df(r, \Delta r) +\epsilon$$
In more general smooth manifolds, we have a smooth function $\varphi:M\rightarrow N$, and the push-forward $\varphi_*$ has the same property in that assigns to each point a linear function which approximates the change in the output of the function $\varphi$ as you vary the input.
But in general smooth manifolds, the tangent space is no longer trivially flat everywhere; the appropriate generalization is that these "small changes" are members of the tangent spaces $TM$ and $TN$ at the appropriate points.
Hence we can think of the push-forward $\varphi_*(x, dx)$ as an assignment of a linear map to each point $x\in M$ which maps small deviations $dx \in TM_{x}$ to deviations $dy\in TN_{\varphi(x)}$ in a linear way. This definition coincides with the usual definition of function differential for real spaces.