---- Let $(M,g)$ be a [[Riemannian manifold]], $D=\nabla$ the [[Levi-Civita connection]] on $M$. The [[Riemannian curvature|Riemann curvature tensor]] $R$ carries a *lot* of data, which motivates a us to extract a simpler 'summary $2$-tensor' via [[tensor of type (p,q)|contraction]] with the [[metric tensor|metric]]. The [[symmetries of the Riemann curvature tensor|symmetries]] of $R=R(X,Y,Z,W)$ mean there's only one way to do this: contracting over the first two or last two indices gives zero; so contract over the inner two (= outer two, again by symmetries). The resulting $2$-tensor is the Ricci curvature of $(M,g)$. > [!definition] ([[Ricci curvature]] - coordinate-free) > With $E_{1},\dots,E_{n}$ an [[orthonormal basis]] of $T_{p}M$, The **Ricci curvature (tensor)** of $g$ is the [[symmetric power|symmetric]][^2] $(0,2)$-[[tensor product of vector bundles|tensor]] $\operatorname{Ric}(X,Y)=\sum_{i={1}}^{n}R(E_{i}, X, Y, E_{i}).$ ^definition (say more about the tensorial aspects) > [!basicproperties] > ^properties - When $n=1$, $R \equiv 0$ and so $\operatorname{Ric}\equiv 0$.[^5] - When $n=2$, $\operatorname{Ric}=\frac{S}{2}g$ for $S=2K$ the scalar curvature; $K$ the [[Gauss curvature]]. (This means [[Gauss curvature]] is intrinsic, the so-called **Theorema Egregium**.) - When $n=3$, have $\operatorname{Ric}_{1 1}=K(E_{1},E_{2})+K(E_{1}, E_{3})$ for $K$ the [[sectional curvature]], and cyclically formulae for $\operatorname{Ric}_{2 2}$ and $\operatorname{Ric}_{3 3}$, assuming [[orthonormal basis|ONB]]. [[symmetries of the Riemann curvature tensor|Indeed]], $\begin{align} \operatorname{Ric}_{1 1}&= \cancel{ R(E_{1},E_{1},E_{1},E_{1}) }^{\text{symmetries}} + \underbrace{ R(E_{2},E_{1},E_{1},E_{2}) }_{ K(E_{1},E_{2}) } +\underbrace{ R(E_{3},E_{1},E_{1},E_{3}) }_{ K(E_{1},E_{3}) }. \end{align}$ Note it follows from this that Einstein implies constant sectional curvature for $3$-manifolds. [^1]: So $\mathbb{S}^{1}$ has zero intrinsic curvature to speak of, only extrinsic as a [[embedded submanifold|submanifold]] of $\mathbb{R}^{2}$. [^2]: Symmetry of $\operatorname{Ric}$ follows from those of the Riemann curvature tensor. Specifically, $\begin{align} \operatorname{Ric}(Y,X) &= \sum_{i=1}^{n} R(E_{i}, Y,X,E_{i}) \\ &= \sum_{i=1}^{n} R(X, E_{i}, E_{i}, Y) \, (\text{inter swap}) \\ &= \sum_{i=1}^{n} R(E_{i}, X, Y, E_{i}) \, (\text{two intra swaps}) \\ &= \operatorname{Ric}(X,Y). \end{align}$ > [!definition] Definition. ([[Ricci curvature]] - coordinate-laden) >The **Ricci curvature (tensor)** 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 $g ^{-1}$. > > 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 > ```