Diffeomorphism ( differential geometry) What is the Geometrical interpretation of diffeomorphism in context of differential geometry ?
 A: You are given two $d$-dimensional manifolds $M$ and $N$. A diffeomorphism $f:\>M\to N$ is in the first place a bijective map. In addition $f$ has to relate the differentiable structures present on $M$ and $N$ in the proper way. This is accomplished if the following holds for all points $p\in M$: 
Let $(x_1,\ldots, x_d)$ be local coordinates in a (suitably small) neighborhood $U$ of $p$, and let $(y_1,\ldots, y_d)$ be local coordinates in the neighborhood $V=f(U)$ of $q:=f(p)$. Then $f\restriction U$ will be described by a function
$\hat f:\ x\mapsto y$ defined in an open set of ${\mathbb R}^d$. I shall omit the hat in the sequel. It is required (a) that  $f$ is at least $C^1$ (often it is required that $f\in C^\infty$), and (b) that $df(x)$ has full rank $d$ at all points of $U$, so that $df^{-1}(y)$ is well defined at all points of $V$.
A diffeomorphism $f$ maps smooth curves $\gamma$ in $M$ to smooth curves in $N$, and its differential $df$ maps a tangent vector to $\gamma$ attached at $p$ to the tangent vector to $f(\gamma)$ attached at $q$.
One more thing: Maybe both $M$ and $N$  are a priori equipped with a Riemannian metric or a distance function. A diffeomorphism does not care about these metrics, nor about angles. It just harmonizes with  the "differentiable structures" on $M$ and $N$.
A: A diffeomorphism on a Riemannian manifold pulls back the metric tensor to another metric tensor.
