I had referred to this structure earlier in a previous question which went unanswered.
If $u$ and $v$ are in $T_pM$ and $n \in (T_pM)^\perp$ then the extrinsic curvature form $K$ be defined as,
$K(u,v) = g(n,\nabla _u V)$
(where $V$ is a local extension of the vector $v$)
Similarly use say $U$ as a local extension of $u$ for defining $K(v,u)$
Then one claims that the following equalities hold,
$K(u,v) = K(v,u) = -g(v,\nabla _u N)$
(where $N$ is a local extension of $n$)
The second equality can hold if one has metric compatible connection and $U(g(N,V)) = V(g(N,U)) = 0$.
I am wondering if $n$ being orthogonal to $u$ and $v$ at $p$ is enough to guarantee the above.
I am unable to understand how the symmetric nature of $K$ can be proven. It doesn't seem to follow just by assuming that the connection is Riemann-Christoffel.