# How is the harmonic map related to the second fundamental form?

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?

## 1 Answer

Here are some references where the authors explain this fact:

Other references that talks about harmonic maps: