Consider a $d$-dimensional complete embedded Riemannian submanifold $(M,g)$ of a Euclidean space $\mathbb{R}^D$. Assume the sectional curvatures of $M$ are bounded within $[\kappa_{\min},\kappa_{\max}]$. Let $p,q\in M$ be two points in $M$ and $v_p\in T_pM$ be a tangent vector. Define the following distance: $$ {\rm dist}(p,q,v_p)=\|\Gamma_{p\to q}v_p-P_{p\to q}v_p\|, $$ where $\Gamma_{p\to q}:T_{p}M\to T_{q}M$ is the parallel transport along the shortest geodesic between $p$ and $q$, and $P_{p\to q}:T_pM\subseteq\mathbb{R}^D\to T_qM\subseteq\mathbb{R}^D$ is the projection to $T_qM$ in the ambient space $\mathbb{R}^D$. In other words, $P_{p\to q}v=P_{T_qM}v$ is the orthogonal projection of $v\in\mathbb{R}^{D}$ to $T_qM$. Notice that $\Gamma_{p\to q}$ is an intrinsic concept while $P_{p\to q}$ is an extrinsic concept.

Our question is whether the following inequality holds: $$ {\rm dist}(p,q,v_p)\leq c_0\|v_p\|d(p,q),\quad \forall p,q\in M,\forall v_p\in T_{p}M, $$ where $c$ is a constant independent of $p,q,$ and $v_p$.

The intuition of this question is to demonstrate that the parallel transport and the projection mapping are close when $p$ and $q$ are close. We expect that this argument holds in a certain uniform sense so the constant $c$ should be independent of $p,q,$ and $v_p$.