How many ways a surface can curve differently in different directions for a n-dimensional embedded submanifolds of $\mathbb{R}^m, m>n$? I think they can curve infinitely many ways but I am not quite certain: don't they have n-dimensional basis? I think I confused something here..
Asked
Active
Viewed 68 times
2
-
2What would you consider different curving? – Hagen von Eitzen Jun 15 '13 at 22:30
-
Different curvature? – 1LiterTears Jun 15 '13 at 22:31
1 Answers
2
Even for a non-umbilic surface in $\mathbb R^3$ you get plane sections with all possible curvatures at $P$ between $k_1$ and $k_2$, where these are the principal curvatures at $P$. So that's infinitely many. For a hypersurface in $\mathbb R^{n+1}$ there are $n$ principal curvatures. We have to decide what you mean by curvature — curvature of curves, curvature of surfaces (that's what sectional curvatures give you), or Gaussian curvature. For higher codimension, we can still talk about sectional curvature or about Gaussian curvature for every normal direction.
Ted Shifrin
- 115,160
-
So you mean for higher dimension and codimension, it is still infinitely many sectional curvature? – 1LiterTears Jun 15 '13 at 23:15
-
2