I recently happened upon a very interesting construction for adapted local additions (used in manifolds of mappings constructions). However, the construction requires a piece of information on parallel transport which I do not get. Here are the details:

Consider a Riemannian manifold $M$ with Riemannian exponential map $\exp_M \colon \Omega_M \rightarrow M$. Assume that $TM = \mathcal{V} \oplus \mathcal{H}$ as smooth vector bundles, where over each $x$, $\mathcal{V}_x$ and $\mathcal{H}_x$ are orthogonal. Now for a smooth curve $\alpha \colon [0,1] \rightarrow M$ and $v \in \mathcal{V}_{\alpha (0)}$, we let $P (\alpha , v) \in \mathcal{V}_{\alpha (1)}$ be the parallel transport in $\mathcal{V}$ induced by the metric over $\alpha$.

For $v \in \Omega_M$ define the smooth curve $\alpha_v \colon [0,1] \rightarrow M,t\mapsto \exp_M (tv)$ and consider the smooth map $$\theta \colon (\mathcal{V} \oplus \mathcal{H}) \cap \Omega_X \rightarrow TM, (v,h) \mapsto P(\alpha_h,v).$$ Then the source claims that for every $x \in M$ the tangent map $T_{0_x} \theta \colon T_{0_x} \Omega_M \rightarrow T_{0_x} T_x M$ is (perhaps up to canonical identifications) the identity. However, I do not see how one would prove this. Actually it would be sufficient for me to see that $T_{0_x} \theta$ is an isomorphism. Does somebody have an idea?