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--- bochner_identity [2022/08/31 18:33] – spencer
+++ bochner_identity [2022/08/31 18:57] (current) – spencer
@@ Line 1: / Line 1: @@
 ====== Bochner Identity ======
-Let $u \in C^2(M,N)$ be a [[harmonic_map|harmonic map]]
+Let $u \in C^2(M,N)$ be a [[harmonic_map|harmonic map]].
+Let $\mathrm{Rm}_N$ denote the Riemann curvature 4-tensor on $N$, $\mathrm{Ric}_M$ the Ricci curvature tensor on $M$, and $\Delta_g$ the Laplace–Beltrami operator on $M$.
+Then the //Bochner identity// is satisfied:
+\[ \Delta_g e(u) = |\nabla (\mathrm{d}u)|^2 + \langle \mathrm{Ric}_M, u^* h\rangle - \mathrm{scal}_g (u^* \mathrm{Rm}_N). \]
+In this formula, $\nabla (\mathrm{d}u) \in \Gamma(T^* M \otimes T^* M \otimes u^*TN)$ is the covariant derivative of the pushforward $\mathrm{d}u$ when one thinks of it as a 1-form with values in the bundle $u^*TN$ equipped with its usual connection (the pullback of the Levi-Civita connection). That is, it is the Hessian of the map $u$.
+Both $\mathrm{Ric}_M, u^*h$ are $(2,0)$-tensors on $M$, and the inner product is the one induced by $g$.
+Finally, $\mathrm{scal}_g (u^*\mathrm{Rm}_N)$ indicates a full trace with $g$ of the pullback of the Riemann curvature tensor on $N$.
+Explicitly, if $\{e_i\}_{i=1}^n$ are an orthonormal basis of $T_p M$ at some point in $M$, then
+\[ (u^*\mathrm{Rm}_N)(e_i,e_j,e_k,e_\ell) = \mathrm{Rm}_N(u_* e_i, u_* e_j, u_* e_k, u_* e_\ell) A_{ijk\ell}, \]
+and
+\[ \mathrm{scal}_g (u^* \mathrm{Rm}_N) = g^{ik} g^{j\ell} R_{ijk\ell}. \]