Let $M$ be a smooth surface and let $x,y \in M$. Let $d_{M}(\cdot, \cdot)$ be the geodesic distance metric on $M$, that is the length of the shortest geodesic curve on $M$. Let $\kappa$ be the maximum principle curvature of all points on $M$ along a minimizing geodesic connecting $x$ and $y$. Let $n_{x}$ and $n_{y}$ be the normal vectors at $x$ and $y$. It is well known result in differential geometry that $\angle(n_x, n_y) \leq \kappa d_{M}(x,y)$.

My question is does an analogous statement hold for higher dimensional manifolds, in particular in the case where the codimension is greater than 1. In the case where codimension is greater than 1, I'm asking if the angle between the normal spaces of two points $x$ and $y$ can be bounded similarly.