Short and elegant definition of the $C^1$ topology

Question

A friend told me that the $\mathbf{C^1}$-topology on the set $C^\infty(M,N)$ of smooth functions between two smooth manifolds $M$ and $N$ can be defined as the coarsest topology making the map $$ C^\infty(M,N) \to C(TM,TN), \qquad f \mapsto df $$ a homeomorphism onto its image, where $C(TM,TN)$ is the space of continuous functions between the tangent bundles $TM$ and $TN$ equipped with the compact-open topology.

Is this definition correct, i.e. equivalent to the usual one by jet spaces? Wikipedia and the other references I found have very long definitions for the C1-topology.

Answer 1

The answer is almost yes if you mean in your post the following: $C^1$-topology should mean the compact open $C^1$-topology and you should replace the $d$ by the mapping

$$T \colon C^1 (M,N) \rightarrow C(M,N) \times C (TM,TN), f \mapsto (f,df)$$ where the factors on the right are endowed with the compact open topology. Then the compact open $C^1$ topology can be defined as the initial topology with respect to $T$ (note this is equivalent to stating that $T$ is a topological embedding onto its closed image (which is what you asked)). Details of this construction can be found in Wockels script https://www.math.uni-hamburg.de/home/wockel/teaching/data/HigherStructures2013/hs.pdf in Section 5 (note that he deals immediately with the compact open $C^\infty$ topology and all proofs deal with this more complicated situation, however all the proofs adapt to switching from $k=\infty$ to $k=1$.

Some comments: For many desirable properties of the compact open topology one would like $M$ to be compact (see Wockel's notes). If $M$ is non-compact there are also different function space topologies, often called Whitney topologies which are often considered and which coincide with the compact open $C^1$-topology if $M$ is compact.