I encounter a definition of $C^\infty$ topology different from the usual one in the book Lectures on hyperbolic geometry.
Def. Let $K$ be a compact subset of $\Bbb H^n$ and let $f,g$ be smooth functions in the neighborhood of $K$ with values in $\Bbb H^n$. Let $P_{y,x}:T_y\Bbb H^n\to T_x\Bbb H^n$ be the parallel transport along the geodesic arc joining $y$ to $x$ and let $d_p$ the $p$-differential. Given $z\in K$ and $p\geq 1$ the mapping $$P_{f(z),z}\circ d_pf(z)-P_{g(z),z}\circ d_pg(z)$$ is a symmetric $p$-linear mapping on the normed space $T_z\Bbb H^n$ and then we can consider its norm. We define the distance between $f$ and $g$ on $K$ by \begin{align*} D(f,g)_K & =\max_{z\in K}d(f(z),g(z))+\\ & +\sum_{p=1}^\infty 2^{-p}(1\wedge\max_{z\in K}\parallel P_{f(z),z}\circ d_pf(z)-P_{g(z),z}\circ d_pg(z)\parallel). \end{align*} A sequence $\{f_i\}$ is said to converge to $f$ w.r.t. the $C^\infty$ topology on $K$ if $D(f_i,f)_K\to 0$.
I guess $\wedge$ means taking the minimum between the left and right sides of $\wedge$. I actually don't understand what $P_{f(z),z}\circ d_pf(z)$ means here. I guess the intention is to take the derivative of $f$ $p$-many times or maybe I'm misunderstanding that it really means taking the derivative $p$-many times ($p$-differential for differential form?).
The real question is that is this definition of $C^\infty$ topology same as the one in Wikipedia?
Comparing this definition with the Whitney $C^k$-topology, they are not exactly the same. The Whitney $C^k$-topology is defined in terms of open sets, while the definition you provided defines the distance between two functions $f$ and $g$ on a compact set $K$. The Whitney $C^k$-topology is a refinement of the $C^\infty$ topology. It considers $C^k$-functions as the basic building blocks and defines a topology based on these functions. On the other hand, the definition you provided seems to define a metric on the set of smooth functions on a compact set $K$.