---- > [!definition] Definition. ([[Ricci curvature]]) > Let $(M,g)$ be a [[Riemannian manifold]], $D$ the [[Levi-Civita connection]] on $M$. The [[Riemannian curvature|Riemann curvature tensor]] carries a *lot* of data; often one would would like to extract simpler quantities. The **Ricci curvature** of $g=g_{ij} dx^{i} \otimes dx^{j}$ at $p$ is the [[bilinear map|bilinear form]] on $T_{p}M$ given by $\begin{align} > \text{Ric}_{p}(\cdot,\cdot): T_{p} M \times T_{p}M &\to \mathbb{R} \\ > (X_{p}, Y_{p} ) & \mapsto \text{tr}\big( R_{p}(X_{p}, -)Y_{p} \big) > \end{align}$ > where $R_{p}(X_{p}, -)$ is the map which takes in a tangent vector $V_{p}$ and returns the [[Riemannian curvature|curvature endomorphism]] $R_{p}(X_{p}, V_{p})$. > > In coordinates, have a [[matrix]] $\text{Ric}=[\text{Ric}_{jk}]_{j,k \in [n]}, \text{where }$ > $\begin{align} > \text{Ric}_{jk} &=\text{Ric}\left( \frac{ \partial }{ \partial x^{j} }, \frac{ \partial }{ \partial x^{k} } \right) \text{, so} \\ > \text{Ric}(X,Y)&= X^{k}Y^{j} \text{Ric}_{jk}. > \end{align}$ > Explicitly, the matrix of $R\left( \frac{ \partial }{ \partial x^{k} }, - \right)\frac{ \partial }{ \partial x^{j} }$ with respect to the basis $\frac{ \partial }{ \partial x^{1} }, \frac{ \partial }{ \partial x^{2} }, \frac{ \partial }{ \partial x^{\ell} }, \dots, \frac{ \partial }{ \partial x^{n} }$ is $[\boldsymbol R_{ \ k \ell} \frac{ \partial }{ \partial x ^{j}}]_{\ell=1,\dots,n}=[\boldsymbol R^{:}_{ j,k \ell}]_{\ell=1,\dots,n}$. Its trace is then the summation $R_{j,k \ell}^{\ell}$. That is, $\text{Ric}_{jk}=R^{\ell}_{j, k \ell}.$ > In terms of the (0,4)-tensor: $\text{Ric}_{jk}=g^{pq}R_{pj, qk}$ > where $(g^{pq})$ are the components of the inverse matrix of $g ^{-1}$. > > It follows from [[symmetries of the Riemann curvature tensor]] (specifically, the third symmetry) that the bilinear form $\text{Ric}$ is [[symmetric multilinear map|symmetric]]. > ---- #### [[trace of a linear operator]] ---- #### 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 > ```