Notes

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====== Harmonic maps ======

Let $(M^n,g), (N,h)$ be closed Riemannian manifolds.
A map $u \in C^2(M,N)$ is said to be //harmonic// if $u$ is a critical point for the [[dirichlet_energy|Dirichlet energy]] functional $E(u)$; equivalently, $\mathrm{d}^* (\mathrm{d}u) = 0$ where $\mathrm{d}u \in \Gamma(T^*M \otimes u^* TN)$ is the pushforward and $\mathrm{d}^* : \Gamma(T^*M \otimes u^*TN) \to \Gamma(u^*TN)$ is the adjoint to the exterior derivative with respect to the $L^2$ metric on sections induced by $g$ and $u^*h$.

===== Examples =====

=== Closed geodesics ===

Let $M = S^1$.
Then a harmonic map $u : M \to N$ is a closed geodesic; that is, a curve that is both a geodesic and periodic.
Indeed, working in a local coordinate $x$ on $S^1$ and coordinates on $N$ so that $u(x) = (u^1(x),\ldots,u^m(x))$, the harmonic map equation becomes that for each $k$,
\[ \frac{\partial^2 u^k}{\partial x^2} - (\Gamma^N)_{ij}^k \frac{\partial u^i}{\partial x} \frac{\partial u^j}{\partial x} = 0, \]
which is precisely the geodesic equation.

=== The identity map ===

The identity $\mathrm{Id} : (M,g) \to (M,g)$ is always harmonic.
Indeed in this case, the pushforward $\mathrm{d}\,\mathrm{Id}$ is the identity map $TM \to TM$ and so is given in a frame $\{e_1, \ldots, e_n\}$ with coframe $\{e^1,\ldots, e^n\}$ by $\mathrm{d}\,\mathrm{Id} = e^i \otimes e_i$.
In particular,
\[ e(\mathrm{d} Id) = n \]
is a constant, **TODO**

===== Second variation formula =====

See (3.38), Eells & Lemaire.

**Claim:** If $u_t$ is a family of harmonic maps, then $\left.\frac{\partial u_t}{\partial t}\right\rvert_{t=0}$ is a Jacobi field $J_u$.

**Proof:** 

We ob