Harmonic maps from the sphere

Harmonic maps from the sphere

In Hamilton's 1997 paper on four-manifolds with positive isotropic
curvature, he considers a local diffeomorphism of Riemannian $n$-manifolds
$$ P: (N,\bar g) \to (M, g). $$ Such a map is harmonic if
$$ \operatorname{tr}_{\bar g} \nabla d P^\alpha = \bar g^{jk} \left(
\frac{\partial P^\alpha}{\partial x^j \partial x^k} +
\Gamma_{\mu\nu}^\alpha \frac{\partial P^\mu}{\partial x^j} \frac{\partial
P^\nu}{\partial x^k} - \bar \Gamma^l_{jk} \frac{\partial
P^\alpha}{\partial x^l} \right) = 0 $$
where $\Gamma$ is the Levi-Civita connection of the pullback metric $ P^*
g $. He then claims that when $(N, \bar g)$ is the round sphere $S^n$,
this is equivalent to $$ \bar g^{jk} ( \bar\Gamma_{jk}^i - \Gamma_{jk}^i )
=0.$$
There must be some property of the connection on the round sphere that
makes this work, but I'm not seeing it - any pointers?
Also, in the first formula for the Laplacian I gave above, $\Gamma$ is
usually interpreted as the pullback connection on $P^* TM$ (which I have
given Greek indices) - this works in more general cases than when $P$ is a
local diffeomorphism. Am I correct that in this case both interpretations
give the same result?