Consider a Riemann manifold $\mathcal{X} \subseteq \mathbb{R}^n$. Given two points $\mathbf{x}, \mathbf{y} \in \mathcal{X}$, denote with $d_g (\mathbf{x}, \mathbf{y})$ the geodesic distance between $\mathbf{x}$ and $\mathbf{y}$. That is, $ d_g (\mathbf{x}, \mathbf{y})$ denotes the length of a geodesic $\gamma\colon [0, 1]\rightarrow \mathcal{X}$ with endpoints $\gamma(0) = \mathbf{x}$ and $\gamma(1) = \mathbf{y}$.
Denote with $\tau \colon \mathcal{X} \rightarrow \mathbb{R}$ a function, such that $\tau(\mathbf{x})$ is the curvature of of the manifold $\mathcal{X}$ at a point $\mathbf{x}$. Can we upper-bound the Euclidean distance $\| \mathbf{x} - \mathbf{y} \|_2$, in terms of $d_g (\mathbf{x}, \mathbf{y})$ and $\tau$?