= Solution
The <vectors> $T=\partial_\tau x$ and $S=\partial_\sigma x$ are coordinate <vector> fields on the parameter surface, so their ordinary commutator vanishes. Torsion-freeness then gives
$$
[T,S]=\nabla_TS-\nabla_ST=0.
$$
Since each $\tau$-curve is geodesic, $\nabla_TT=0$. Apply the Ricci identity to $V=T$ and use $\nabla_ST=\nabla_TS$:
$$
(\nabla_T\nabla_TS)^\alpha
=R^\alpha{}_{\beta\mu\nu}T^\beta T^\mu S^\nu.
$$
Thus the desired form holds with
$$
\boxed{E^\alpha{}_{\nu}=-R^\alpha{}_{\beta\mu\nu}T^\beta T^\mu.}
$$
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