# connections on an embedded sumanifolds of Euclidean space

I would like to see how to derive the connection 1 form on the bundle of oriented orthonormal frames for a manifold embedded in Euclidean space.

It seems that the geometry resides in a collection of 1 forms on the frame bundle the arise naturally from the embedding. Given a frame (e1,...,en) over a point in the manifold, map it to e1. This is a smooth map of the frame bundle into Euclidean space. The 1 forms are then defined as

wi = de1.ei 1=2,...n

The action of the rotation group on the frame bundle leaves these forms invariant because it preserves the Euclidean inner product. That is

dL(e1).L(ei) = Ld(e1).L(ei) = de1.ei

It is tempting to think that these 1 forms are the components of a connection 1 form but I do not not see how to formalize it.

No, you're confusing the frame bundle and the sphere bundle. For the $\mathfrak{so}(n)$-valued $1$-form on the frame bundle, you need $$\omega_{ij} = de_i\cdot e_j\,, \quad 1\le i,j \le n\,.$$
• Yes. And to get the extrinsic geometry — the second fundamental form — you might want an orthonormal basis for the normal space, too, $e_{n+1},\dots,e_N$ (with $M\subset\mathbb R^N$). May 12, 2013 at 2:42
• Well, the matrix $[\omega_{ij}]$ is skew-symmetric, so it's giving you an $\mathfrak{so}(n)$-valued $1$-form. When you change frame by $a\colon M\to SO(n)$, it transforms by $\omega \rightsquigarrow a\omega a^{-1} + da\cdot a^{-1}$, which is what it should. [You should come take my graduate Riemannian geometry course this fall! :)] May 19, 2013 at 17:58