Let $\mathcal{M}$ be a smooth Riemannian manifold embedded in Euclidean space. We used the Euclidean metric as Riemannian metric. $T_x\mathcal{M}$ denote the tangent space of $\mathcal{M}$ at $x$, and $P_{T_x\mathcal{M}}(\cdot)$ denote the projection operator on tangent space.
My question is whether there exits constant $\delta$ such that $$ \|P_{T_x\mathcal{M}}(w) - \mathcal{T}_{y,x}P_{T_y\mathcal{M}}(w)\| \leq \delta \|\mbox{exp}_x^{-1}y\|\|w\| $$ for any $x,y,w$. In which $\mbox{exp}^{-1}$ denote the inverse exponential mapping and $\mathcal{T}_{y,x}$ denote the parallel transport from $T_y\mathcal{M}$ to $T_x\mathcal{M}$ .