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

Question

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?

Answer 1

Here are some references where the authors explain this fact:

• Fanghua Lin & Changyou Wang's The Analysis of Harmonic Maps and Their Heat Flows (pages 2 and 3)
• Struwe's chapter on Nonlinear partial differential equations in differential geometry (page 262).

Other references that talks about harmonic maps:

• Frédéric Hélein and John C. Wood's chapter (#8) from Handbook of Global Analysis
• Jürgen Jost book Riemannian Geometry and Geometric Analysis chapter 9.
• Richard Schoen and Shing-Tung Yau's book Lectures on Harmonic Maps