----- Warning re: sign conventions is probably wise. > [!proposition] Proposition. ([[symmetries of the Riemann curvature tensor]]) > Let $(M,g)$ be a [[Riemannian manifold]] of dimension $n$. Consider the [[Riemannian curvature|Riemann curvature tensor]] $R$. It has the following symmetries in its indices $i,j,k,\ell$: > > 1. **(Intra-comma skew-swaps)** $R_{ij,k\ell}=-R_{ij, \ell k}=R_{ji, k \ell}$ ; > 2. **(1st Bianchi identity)** $R^{i}_{j, k \ell}+R^{i}_{\ell, jk}+R^{i}_{k, \ell j}=0$ ; > 1. Equivalently: $R_{ij, k\ell}+R_{ik, \ell j}+R_{i \ell, j k}=0$; > 3. **(Inter-comma swaps)** $R_{ij, k \ell}=R_{k \ell, ij}$. > > - [ ] where did the coordinate-free version go? is it in a different note? > [!proof]- Proof. ([[symmetries of the Riemann curvature tensor]]) > ~ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```