I'm trying to understand the following:
Let $C$ be a $n$ dimensional minimal cone in $\mathbb R^{n+1}$ with vertex at $0$ with second fundamental form $h_{ij} = h_{ji}$. (Minimal implies $\sum_i h_{ii} = 0$)
Also, let $h_{ijk}$ be the components of the one-form definied by $$\sum_k h_{ijk} \omega_k = \mathrm d h_{ij} - \sum_k h_{ik} \, \omega_{kj} - \sum_k h_{jk} \, \omega_{ki},$$ where all sums are from $1$ to $n$, $\{\omega_1,\cdots,\omega_n\}$ are the dual frames to $\{e_1,\cdots,e_n\}$ and $\{\omega_{ij}\}_{1\leq i,j \leq n}$ are the connection one-forms.
Choose a frame $\{e_1,\cdots,e_n\}$ such that $h_{ij}$ is diagonal and $e_n$ is in radial direction (ie. $e_n = x/|x|$).
Then we have $h_{ij} = h_{nn} = 0$, $i \neq j$ and $h_{ijn} = - |x|^{-1} \, h_{ij}$, $i,j = 1,\cdots,n$
Now I understand that $h_{nn} = 0$ since in the radial direction, the cone just looks like a straight line.
My Question: How do I see the second identity? $$h_{ijn} = - |x|^{-1} \, h_{ij}, \quad i,j = 1,\cdots,n$$
EDIT So my idea was to look at this \begin{align} h_{ijn} & = (\mathrm d h_{ij})(e_n) - \sum_k h_{ik} \, \omega_{kj}(e_n) - \sum_k h_{jk} \, \omega_{ki}(e_n) \\ % & = \partial_n h_{ii} \, \delta_{ij} - \Gamma^i_{nj} \, h_{ii} - \Gamma^j_{ni} \, h_{jj} \end{align}
Any suggestions?
Since a cone is radially self-similar, and curvature (as measured by the second fundamental form) scales like inverse length, its curvatures at $x$ and $\lambda x$ (for any $\lambda > 0$) will be related by $$h_{ij}(\lambda x) = \lambda^{-1} h_{ij}(x)$$ if we work in a frame that is parallel in the radial direction. That is, in this frame, the components of the second fundamental form are positively homogeneous of degree $-1$.
In terms of connection coefficients, this extra condition on the frame translates to $\omega_{ijn}=0;$ so we have the simple formula $h_{ijn} = \partial_n h_{ij}.$ By Euler's homogeneous function theorem, we have the derivative formula $$x^k \partial_k h_{ij} = |x| h_{ijn}= -h_{ij},$$ which can be rearranged to give $h_{ijn} = -|x|^{-1} h_{ij}$ as desired.
Finally, to relax the extra condition on the frame I added, note that your $h_{ijn}$ is really the covariant derivative $\nabla_n h_{ij}$ written in terms of moving frames; so we have shown the coordinate-independent equation $\nabla_X h = -|x|^{-1} h$ for $X$ the radial unit vector field, and thus $h_{ijn} = \nabla_n h_{ij} = -|x|^{-1} h_{ij}$ whenever $e_n =X$.