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--- harmonic_map [2022/08/31 15:17] – spencer
+++ harmonic_map [2022/09/01 13:05] (current) – spencer
@@ Line 13: / Line 13: @@
 \[ \frac{\partial^2 u^k}{\partial x^2} - (\Gamma^N)_{ij}^k \frac{\partial u^i}{\partial x} \frac{\partial u^j}{\partial x} = 0, \]
 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