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I am reading this paper that has the following:

Suppose $M$ is an (n-1) dimensional closed hyper surface immersed in $\mathbb{R}^{n}$. Let $e_1, \cdots, e_n$ be orthonormal frame in $\mathbb{R}^n$ such that $e_1,\cdots, e_{n-1}$ are tangent to $M$ and $e_n$ is the outer normal. Let $\omega_i$ be the corresponding coframes and $\omega_{i,j}$ be the connection forms. And use the same notation for the pull back of the forms through immersion. Then the second fundamental form is defined by the symmetric matrix $\{h_{ij}\}$ with $$\omega_{i,n+1}=h_{ij}\omega_j.$$

I use Boothby's An introduction to differentiable manifolds and Riemannian geometry for reference on differential geometry. In this book, the notations are like this:

  1. $e_1,\cdots,e_n$ are the orthonormal frames.
  2. $\omega^i$ are the coframes. (so I figured this is equal to $\omega_i$ in the paper.)
  3. $\omega_{j}^{k}$ are defined so that $\nabla_{X} e_j = \Sigma_k \omega_{j}^{k}(X)e_k$, where $X$ is a vector field.
  4. $\omega_{i,j}=\Sigma_k \omega_{i}^kg_{kj}$. In this case, since this is an orthonormal frame, I guess $\omega_{i,j}=\omega_i^j$.

I assume the symmetric matrix is defined by $h_{ij}=\langle e_n,\nabla_{e_i}e_j\rangle$. (Correct me if I am wrong.) Using 3 and 4, I get $h_{ij}=\omega_{j,n+1}(e_i)$. To arrive at the equation in the paper, I will have to justify (note that $\omega_j=\omega^j$) $$\Sigma_j \omega_{j,n+1}(e_i)\omega_j = \omega_{i,n+1}.$$

I am stuck at this step. Can anyone help me?

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  • $\begingroup$ for curiosity, which paper were you reading? $\endgroup$ – cesare borgia Nov 8 '17 at 17:40
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Basically, you have to prove that $\omega_{i, n}(e_l) = \omega_{l, n}(e_i)$, which is proved from $\omega^k_l(e_i) = \omega^k_i(e_l)$. Now, see that $\nabla_{e_l}e_i = \sum_k\Gamma^k_{li}e_k$, the usual Christoffel symbols. Seeing that the Christoffel symbols are symmetric (see Christoffel symbols and fundamental forms, for example), what you want follows.

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  • $\begingroup$ I guess what the author meant by "orthonormal frame" is orthonormal list of vector fields induced from local parametrization. $\endgroup$ – henryforever14 Jun 30 '15 at 14:31

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