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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.

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  • $\begingroup$ 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... $\endgroup$ Aug 22, 2013 at 23:32

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