# 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?

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In the second equation you quoted, Hamilton is in effect redefining $g$ to be the pullback metric $P^*g$ on $S^n$, and $P$ to be the identity map of $S^n$. So in this case the Jacobian of $P$ is the identity matrix, and its second partial derivatives all vanish. It has nothing to do with whether the manifold is the round sphere or not; it's just what the harmonic map equation reduces to when you apply it to the identity map between two metrics on the same manifold.
Thankyou very much, it seems obvious in retrospect. I guess the wording "or equivalently on $S^n$" led me too far down that path without stopping to think it through properly... – Anthony Carapetis Aug 22 '13 at 23:32