How many connections in \(TM\) exist?

The answer is given in [@lee2013smooth_manifolds, Theorem 4.14], which states for a given connection \(\nabla^0\) in \(TM\) (\(M\) is a smooth manifold with or without boundary) the set of all connections is given by

\begin{equation*} \mathcal{A}(TM)=\{\nabla^0+D: D\in \Gamma(T^{(1,2)}TM)\}, \end{equation*}

where \(\Gamma(T^{(1,2)})\) denotes the set of smooth \((1,2)\)-tensor fields .

Furthermore, every [smooth manifold](smooth manifold.md) with or without boundary admits a connection in \(TM\) [@lee2018riemannian_manifolds, Proposition 4.12].