# Normal Variation on Manifolds

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.

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.
• This is great, thank you for your answer! An additional question. For 2-manifolds with codimension > 1 is the Lipschitz constant still bounded by the maximum principle curvature? And then for m-manifolds in $\mathbb{R}^n$ is the Lipschitz constant bounded by some function of the sectional curvature? – user62013 Nov 23 '14 at 20:30
• @user62013 Well, the "principle curvatures" are not defined if codimension $>1$. – Anton Petrunin Nov 24 '14 at 4:21
• While the principle curvatures may not be defined, it's not immediately obvious that one could not define something similar in the general case. If I consider a surface $M$ in $\mathbb{R}^3$ then principal curvature is defined at a point in terms the intersection with a plane which gives a curve $\gamma$. If I embed $M$ in a higher dimension it still has the same curvature at that point and still has the same $\gamma$ intersecting at $p$. Perhaps one can't intersect it with a plane but perhaps a hyperplane. – user62013 Nov 24 '14 at 5:15
• OK, you want to bound the Lipschitz constant of the Gauss map in terms of normal curvatures. This is a question in linear algebra. Set $q(x,y,z,w)=\langle s(x,y),s(z,w)\rangle$ then you get all the values $q(x,x,x,x)$ and you need to estimate $\sum q(u,e_i,u,e_i)$, but I am too lazy. – Anton Petrunin Nov 24 '14 at 17:34