Riemannian geometry: ...Why is it called 'Exponential' map? The exponential map $exp_{p}:T_{p}M \to M$ given a suitable $v \in T_{p}M$, returns $p$, displaced along the geodesic uniquely determined by $(p,v) \in TM$ for unit "time".  
So, what does the above have to do with the familiar concept/s of exponentiation?
Why does this map carry this name?  

The function $f(t)=\mathrm{e}^{at}$ can be viewed as the solution of the initial value problem:
$$
x'=ax, \quad x(0)=1.
$$
More generally, if $A\in\mathbb R^{n\times n}$ and $u_0\in\mathbb R^n$, then the solution of 
the
initial value problem:
$$
x'=Ax, \quad x(0)=u_0,
$$
is $u(t)=\mathrm{e}^{tA}u_0$, where $\mathrm{e}^{tA}$ is the exponential of the matrix $A$. It $A$ possessed imaginary eigenvalues, then $u(t)$ returns to a multiple of itself, for suitable initial $u_0$.
