I want to prove Gauss' Theorem:
"Let p $\in$ M and x,y orthonormal vectors of $T_p M$. So k(x,y)-$\bar{k}$(x,y)= < B(x,x),B(y,y)> - |B(x,y)|$^2$.
From Riemannian Geometry, Manfredo do Carmo
We have:
$X,Y \in \Gamma (TU)$ where $U$ is a neighborhood in $M$
$\bar{X},\bar{Y}$ are extensions of $X,Y$ in $\bar{M}$
$\nabla$ is a Riemannian connection in $M$
$\bar{\nabla}$ is a Riemannian connection in $\bar{M}$
$H_\eta(x,y)= < B(x,y),\eta >, x,y\in T_p M$
$B(X,Y)=\bar{\nabla}_{\bar{X}} \bar{Y} - \nabla _{X} Y$
I'm stuck in this step:
$< \bar{\nabla} _ \bar{Y} \bar{\nabla}_\bar{X} \bar{X},Y> = \sum_i < H_i(X,X)\bar{\nabla}_\bar{Y} E_i + \bar{Y} H_i (X,X) E_i, Y> + <\bar{\nabla} _ \bar{Y} \nabla_X X,Y> $
And i have to prove that this is:
$< \bar{\nabla} _ \bar{Y} \bar{\nabla}_\bar{X} \bar{X},Y> = - \sum_i H_i(X,X)H_i (Y,Y) + <\bar{\nabla} _ \bar{Y} \nabla_X X,Y> $
Can someone help with a hint of how to go to this step?
I think i can do $<\bar{Y} H_i (X,X) E_i, Y>=0$ because $E_i$ and $Y$ are orthogonal.