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For a smooth $f $ between (pseudo-)Riemannian manifolds $M$ and $N$, the harmonic maps equation (or the wave maps equation) is: $$ \text{trace} \nabla \text d f = 0 $$ where $\nabla$ is the Levi-Civita connection induced on $T^*M \otimes f^* TN$ and $\text d f = \frac {\partial f^i}{\partial x^\alpha} \text d x ^\alpha \otimes \frac{\partial}{\partial f^i}$ is a section of $T^* M \otimes f^* TN$. If we write out the equation in coordinates, we see that $$ \Delta_g f = - g(\text{grad} f^\alpha, \text{grad}f^\beta) {}^h \Gamma_{\alpha \beta}^i \partial_i $$ where ${}^h \Gamma_{\alpha \beta}^i$ is the Christoffel symbol on $N$. I have seen the claim that if $N$ is an immersed submanifold of Euclidean space, then the harmonic maps equation becomes $$ \Delta_g f = - \text{II}(\nabla_a f, \nabla^a f) $$ where $\text{II}$ is the second fundamental form of $N$. But I cannot see how this would follow from the previous expression. I also suspect this holds so long as $N$ is an immersed submanifold of some larger ambient manifold, so I believe the proof could possibly proceed without appealing to coordinates. Is there a thorough reference that explains this connection?

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Here are some references where the authors explain this fact:

Other references that talks about harmonic maps:

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