= Wave map
{title2=$\Box\phi=\phi(|\phi_t|^2-|\nabla\phi|^2)$}
A wave map is a <harmonic map> with Lorentzian domain: a critical point of the action obtained by contracting the pullback target metric with the domain <Lorentzian metric>. For target <sphere> $S^2$ in Euclidean space and $\Box=-\partial_t^2+\Delta$, its extrinsic equation is $\Box\phi=\phi(|\phi_t|^2-|\nabla\phi|^2)$, subject to $|\phi|=1$. The derivative contractions are <null forms for wave equations>. <Wave map Cauchy data> must include a tangent initial velocity.
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