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I have found this identity in a paper that I am reading:

On a Riemannian $n$-manifold $(M,g)$, consider a normal coordinate system $\{x^i\}_{i=1,\dots,n}$centered at $p\in M$. Then $$g_{a b, c d}(p) + g_{a d,b c}(p) + g_{a c, d b}(p) = 0$$ Where $g_{a b, c d} = \frac{\partial^2 g(\frac \partial {\partial x^a }, \frac \partial {\partial x^b })} {\partial {x^d}\partial {x^c}}. $ How can I prove this statement?

I recall that in a normal coordinate system centered at $p$ the following identities hold (notice that are true only at $p$): $$ g_{i j}(p) = \delta_{i j} \text{ (Kronecker delta)}$$ $$ \partial_{x^k} g_{i j}(p) = 0 $$ $$ \Gamma_{i j}^k (p) = 0$$ $$\partial_{x^k}\Gamma_{i j}^l(p) = \frac 1 2 (g_{i l,j k}(p) + g_{j l,i k}(p)- g_{i j, l k}(p))$$ Where $\Gamma_{i,j}^k$ are the Christoffel symbols of the Levi-Civita connection induced by $g$.

So we have that (at the point $p$) $$g_{a b,c d} + g_{a d, b c} + g_{a c,d b} = 2 \partial_{x^d} \Gamma_{b c}^a + g_{b c, a d} + g_{a c,d b} = 2 \partial_{x^d} \Gamma_{b c}^a+ 2 \partial_{x^d} \Gamma_{b a}^c + g_{a b, c d} $$

The fourth equation seems to be promising but still I don't see how to conclude.

Overflowian
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1 Answers1

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The Gauss Lemma can be stated as $$\sum_i x^iu^i = \sum _ix^ig_{i j}(x) u^j \quad \forall u \in \mathbb{R}^n$$ in normal coordinates $\{x^i\}_{i=1,\dots, n}$. In particular we have that $$1)\quad \quad x^j = x^ig_{ij}(x).$$ Deriving 1) along $\partial_k$ we obtain $$\partial_k x^j = \delta_k^j = \partial_k x^i g_{i j}(x) + x^i g_{i j,k}(x) = g_{k j}(x) + x^i g_{i j,k}(x)$$ Deriving again along $\partial_r$ we obtain $$ 0 = g_{k j, r}(x) + g_{r j, k}(x) + x^i g_{i j,k r}(x) $$ Deriving again along $\partial_l$ we obtain $$0 = g_{k j, r l}(x) + g_{r j, k l}(x) + g_{l j,k r}(x) + x^i g_{l j,k r l}(x)$$ Now we evaluate at the origin this last equation to get $$0 = g_{k j, r l}(O) + g_{r j, k l}(O) + g_{l j,k r}(O) = g_{j k, r l}(O) + g_{j r, l k}(O) + g_{ j l,k r}(O) $$ which is the wanted equation.

Overflowian
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