Why does the condition $\langle \varphi _k\left( t \right) |\dot{\varphi}_l\left( t \right) \rangle =0$ called parallel transport condition?

I'm reading this paper about nonadiabatic holonomic quantum computation and met the condition eq.(2) states that $$\langle \varphi _k\left( t \right) |\dot{\varphi}_l\left( t \right) \rangle =0$$ is the parallel transport condition, where $$|\varphi _k\left( t \right) \rangle$$ means quantum state at time $$t$$ and is the $$k$$-th eigenstate of the evolution Hamiltonian. I know that parallel transport means moving vectors in manifold parallelly, but I do not quite understand why this formula have the name, and the paper didn't explain it.

More precisely, they look at the Schrödinger time evolution of a $$L$$-dimensional subspace under the assumption that it is periodic with period $$\tau$$. Then, given a basis of this subspace, one can ask how much is this basis rotated after time $$\tau$$? This is what the concept of holonomy is about and can be quantified by introducing an appropriate connection. Since there's a freedom in choosing the basis, there's a $$U(L)$$ "gauge freedom". Technically, this means that the connection is a principal connection on a unitary frame bundle of rank $$L$$. The stated condition is exactly the condition that the time evolution has to fulfill to yield a parallel transport w.r.t. this connection. I suggest to look at previous works, Refs. [13,14] in that paper, to understand that better.