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. 



Normal Variation on Manifolds
The maximum principle curvature is the upper bound for the Lipschitz constant of the Gauss map at the point.
Integrating, you get your estimate.


The same can be done in higher dimensions. 
If $M$ is $m$-dimensional submanifold in $\mathbb R^n$,
you have Gauss map $\nu\colon M\to \mathrm{Gr}(m,n)$,
where $\mathrm{Gr}(m,n)$ is the Grassmannian of $m$-subspaces of $\mathbb R^n$.
The Lipschitz constant of $\nu$ at given point $p$
can be calculated in terms of second fundamental form\* $s$:
$$\mathrm{lip}\_p\nu=\sup\_{u\in T\_p, |u|=1}\left\{\sqrt{\sum\_i|s(e\_i,u)|^2}\right\},$$
where $e\_i$ is an orthonormal basis in in the tangent space $T\_p$.


(\*)The second fundamental form $s$ is a quadratic form on the tangent bundle with values in the normal bundle defined as 
$$s(v,w)=(\nabla\_v w)^\bot,$$
where $(\nabla\_v w)^\bot$ denotes the orthogonal projection of covariant derivative $\nabla\_v w$ onto the normal bundle.


