Suppose that $v$ and $w$ are orthogonal unit vectors in $T_p\Sigma.$ Show that $\kappa_p(v)+\kappa_p(w)$ is independent of the specific choice of $v$ and $w$ as long as they are orthogonal.
$T_p\Sigma$ is the tangent plane of a surface $\Sigma$ at a point $p$.
How can I prove this? Do I have to use the second fundamental form to show this or is there another straight forward way?
If $v$ and $w$ are orthogonal unit vectors of $T_p\Sigma$ then wouldn't $\kappa_p(v)$ just be the normal curvature? If it is the normal curvature then we can define it by taking an arc-length parametrized curve $\alpha$ in $\Sigma$ such that $\alpha(0)=p$ and $\alpha'(0)=v$ which passes through the point $p$ and has $v$ as its tangent vector at the point $p$. But then wouldn't the value of the normal curvature of $\alpha$ depend only on the unit tangent vector $v$?