[proofplan] We first compute the time variation of the Christoffel symbols under Ricci flow and use it to express $\partial_t R_{ijk\ell}$ in terms of second covariant derivatives of the Ricci tensor, together with the contribution from differentiating the lowering metric. We then prove the differential Bianchi commutation identity that rewrites precisely that second-Ricci expression as the rough Laplacian of the Riemann tensor plus the required quadratic curvature terms and Ricci-linear terms. Substituting this identity into the variation formula gives the stated evolution equation for the fully covariant curvature tensor. [/proofplan]

[step:Compute the Ricci flow variation of the Christoffel symbols]Fix $t \in I$ and work in a local coordinate chart $(U,\varphi)$ with coordinates $(x_1,\dots,x_n)$. Let $h_{ij}$ denote the metric variation, defined by \begin{align*} h_{ij} := \partial_t g_{ij}. \end{align*} Since $\nabla g = 0$, differentiating the identity defining the Levi-Civita connection gives \begin{align*} \partial_t \Gamma_{ij}^p = \frac{1}{2}g^{pq} \left( \nabla_i h_{jq} + \nabla_j h_{iq} - \nabla_q h_{ij} \right). \end{align*} Along Ricci flow, $h_{ij} = -2\operatorname{Ric}_{ij}$, hence \begin{align*} \partial_t \Gamma_{ij}^p = - g^{pq} \left( \nabla_i \operatorname{Ric}_{jq} + \nabla_j \operatorname{Ric}_{iq} - \nabla_q \operatorname{Ric}_{ij} \right). \end{align*}[/step]

[guided]We first isolate the part of the calculation that depends only on the metric variation. Define the symmetric tensor $h_{ij}$ by \begin{align*} h_{ij} := \partial_t g_{ij}. \end{align*} The Christoffel symbols are \begin{align*} \Gamma_{ij}^p = \frac{1}{2}g^{pq} \left( \partial_{x_i}g_{jq} + \partial_{x_j}g_{iq} - \partial_{x_q}g_{ij} \right). \end{align*} Although this formula uses partial derivatives, the variation of $\Gamma_{ij}^p$ is tensorial in the lower indices once the upper index is fixed, and the standard covariant form is obtained by replacing partial derivatives of $h$ by covariant derivatives. Since the Levi-Civita connection satisfies $\nabla g = 0$, differentiating the displayed Christoffel formula and collecting the connection terms gives \begin{align*} \partial_t \Gamma_{ij}^p = \frac{1}{2}g^{pq} \left( \nabla_i h_{jq} + \nabla_j h_{iq} - \nabla_q h_{ij} \right). \end{align*} Now insert the Ricci flow equation. By definition of Ricci flow, \begin{align*} h_{ij}=\partial_t g_{ij}=-2\operatorname{Ric}_{ij}. \end{align*} Therefore \begin{align*} \partial_t \Gamma_{ij}^p = \frac{1}{2}g^{pq} \left( -2\nabla_i \operatorname{Ric}_{jq} -2\nabla_j \operatorname{Ric}_{iq} +2\nabla_q \operatorname{Ric}_{ij} \right). \end{align*} Equivalently, \begin{align*} \partial_t \Gamma_{ij}^p = - g^{pq} \left( \nabla_i \operatorname{Ric}_{jq} + \nabla_j \operatorname{Ric}_{iq} - \nabla_q \operatorname{Ric}_{ij} \right). \end{align*} This formula is the input for differentiating curvature: it converts the time derivative of the connection into first covariant derivatives of Ricci.[/guided]

[step:Differentiate the curvature tensor and lower the final index]Define the mixed curvature tensor by \begin{align*} R_{ijk}{}^p := \partial_{x_i}\Gamma_{jk}^p - \partial_{x_j}\Gamma_{ik}^p + \Gamma_{iq}^p\Gamma_{jk}^q - \Gamma_{jq}^p\Gamma_{ik}^q. \end{align*} The standard curvature-variation identity follows by differentiating this coordinate formula and replacing the resulting partial derivatives of $\partial_t\Gamma$ by covariant derivatives, since $\partial_t\Gamma$ is a $(1,2)$-tensor: \begin{align*} \partial_t R_{ijk}{}^p = \nabla_i(\partial_t\Gamma_{jk}^p) - \nabla_j(\partial_t\Gamma_{ik}^p). \end{align*} Lowering the final index by $R_{ijk\ell}=g_{\ell p}R_{ijk}{}^p$ and using $\partial_t g_{\ell p}=-2\operatorname{Ric}_{\ell p}$ yields \begin{align*} \partial_t R_{ijk\ell} = -2\operatorname{Ric}_{\ell p}R_{ijkp} + g_{\ell p}\partial_t R_{ijk}{}^p. \end{align*} Substituting the formula for $\partial_t\Gamma_{ij}^p$ and using $\nabla g=0$ gives \begin{align*} \partial_t R_{ijk\ell} &= -\nabla_i\nabla_j\operatorname{Ric}_{k\ell} -\nabla_i\nabla_k\operatorname{Ric}_{j\ell} +\nabla_i\nabla_\ell\operatorname{Ric}_{jk} +\nabla_j\nabla_i\operatorname{Ric}_{k\ell} +\nabla_j\nabla_k\operatorname{Ric}_{i\ell} -\nabla_j\nabla_\ell\operatorname{Ric}_{ik} -2\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} We commute the first and fourth second derivatives of the Ricci tensor. For a covariant $2$-tensor $B_{k\ell}$, the curvature commutator is \begin{align*} (\nabla_j\nabla_i-\nabla_i\nabla_j)B_{k\ell} = -R_{jikp}B_{p\ell}-R_{ji\ell p}B_{kp}. \end{align*} Applying this to $B_{k\ell}=\operatorname{Ric}_{k\ell}$ gives the complete second-Ricci variation formula \begin{align*} \partial_t R_{ijk\ell} &= -\nabla_i\nabla_k\operatorname{Ric}_{j\ell} +\nabla_i\nabla_\ell\operatorname{Ric}_{jk} +\nabla_j\nabla_k\operatorname{Ric}_{i\ell} -\nabla_j\nabla_\ell\operatorname{Ric}_{ik} -R_{jikp}\operatorname{Ric}_{p\ell} -R_{ji\ell p}\operatorname{Ric}_{kp} -2\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*}[/step]

[guided]We must be careful here because differentiating the connection produces two second derivatives of $\operatorname{Ric}_{k\ell}$ with reversed order. Those terms do not cancel unless the commutator is included. Define the mixed curvature tensor by \begin{align*} R_{ijk}{}^p := \partial_{x_i}\Gamma_{jk}^p - \partial_{x_j}\Gamma_{ik}^p + \Gamma_{iq}^p\Gamma_{jk}^q - \Gamma_{jq}^p\Gamma_{ik}^q. \end{align*} Let $s \in \mathbb{R}$ be a parameter with $t+s \in I$. Because $\partial_t\Gamma$ is the first-order variation at $s=0$ of the difference between the Levi-Civita connection of $g(t+s)$ and the Levi-Civita connection of $g(t)$, it is tensorial. Therefore the coordinate differentiation of curvature can be rewritten covariantly as \begin{align*} \partial_t R_{ijk}{}^p = \nabla_i(\partial_t\Gamma_{jk}^p) - \nabla_j(\partial_t\Gamma_{ik}^p). \end{align*} Lowering the last index gives one additional term because the metric itself evolves: \begin{align*} \partial_t R_{ijk\ell} = (\partial_t g_{\ell p})R_{ijk}{}^p +g_{\ell p}\partial_t R_{ijk}{}^p = -2\operatorname{Ric}_{\ell p}R_{ijkp} +g_{\ell p}\partial_t R_{ijk}{}^p. \end{align*} Now substitute \begin{align*} \partial_t \Gamma_{ab}^p = -g^{pq}(\nabla_a\operatorname{Ric}_{bq}+\nabla_b\operatorname{Ric}_{aq}-\nabla_q\operatorname{Ric}_{ab}). \end{align*} Since $\nabla g=0$, the metric used to raise the index is parallel and can be moved through covariant derivatives. This gives \begin{align*} \partial_t R_{ijk\ell} &= -\nabla_i\nabla_j\operatorname{Ric}_{k\ell} -\nabla_i\nabla_k\operatorname{Ric}_{j\ell} +\nabla_i\nabla_\ell\operatorname{Ric}_{jk} +\nabla_j\nabla_i\operatorname{Ric}_{k\ell} +\nabla_j\nabla_k\operatorname{Ric}_{i\ell} -\nabla_j\nabla_\ell\operatorname{Ric}_{ik} -2\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} The first and fourth terms are not equal in general. We use the covariant-derivative commutator for a covariant $2$-tensor $B_{k\ell}$: \begin{align*} (\nabla_j\nabla_i-\nabla_i\nabla_j)B_{k\ell} = -R_{jikp}B_{p\ell}-R_{ji\ell p}B_{kp}. \end{align*} With $B_{k\ell}=\operatorname{Ric}_{k\ell}$, this yields \begin{align*} \partial_t R_{ijk\ell} &= -\nabla_i\nabla_k\operatorname{Ric}_{j\ell} +\nabla_i\nabla_\ell\operatorname{Ric}_{jk} +\nabla_j\nabla_k\operatorname{Ric}_{i\ell} -\nabla_j\nabla_\ell\operatorname{Ric}_{ik} -R_{jikp}\operatorname{Ric}_{p\ell} -R_{ji\ell p}\operatorname{Ric}_{kp} -2\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} This is the exact place where the suppressed commutator terms enter the proof.[/guided]

[step:Derive the Bianchi commutation identity with the stated curvature convention]We use the following sign-sensitive commutation identity, valid for the curvature convention in the statement and the Ricci contraction $\operatorname{Ric}_{ij}=R_{pij p}$: \begin{align*} -\nabla_i\nabla_k\operatorname{Ric}_{j\ell} +\nabla_i\nabla_\ell\operatorname{Ric}_{jk} +\nabla_j\nabla_k\operatorname{Ric}_{i\ell} -\nabla_j\nabla_\ell\operatorname{Ric}_{ik} -R_{jikp}\operatorname{Ric}_{p\ell} -R_{ji\ell p}\operatorname{Ric}_{kp} = \Delta R_{ijk\ell} + 2\left( R_{ipkq}R_{jp\ell q} - R_{ip\ell q}R_{jpkq} \right) - R_{ijpq}R_{k\ell pq} - \operatorname{Ric}_{ip}R_{pjk\ell} - \operatorname{Ric}_{jp}R_{ipk\ell} - \operatorname{Ric}_{kp}R_{ijp\ell} + \operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} [claim:Bianchi commutation identity for the curvature Laplacian] For the curvature convention \begin{align*} R_{ijk\ell}=g_{\ell p}R_{ijk}^{\ \ \ p}, \end{align*} the identity displayed above holds. [/claim] [proof] Fix a point $x_0 \in U$ and choose [normal coordinates](/theorems/2713) for $g(t)$ at $x_0$, so that $g_{ij}(x_0)=\delta_{ij}$ and $\Gamma_{ij}^p(x_0)=0$. Since every term in the identity is tensorial, it suffices to verify the identity at $x_0$. The [second Bianchi identity](/theorems/1541) in this curvature convention is \begin{align*} \nabla_p R_{ijk\ell} + \nabla_i R_{jpk\ell} + \nabla_j R_{pik\ell} = 0. \end{align*} Apply $\nabla_q$ to the second Bianchi identity and contract the differentiated index $q$ with the index $p$ using $g^{pq}$. Since $\nabla g=0$, the contraction may be moved through the covariant derivative. At $x_0$ this gives \begin{align*} \Delta R_{ijk\ell} = -\nabla_p\nabla_i R_{jpk\ell} - \nabla_p\nabla_j R_{pik\ell}. \end{align*} Commute the covariant derivatives in each term. For a covariant $4$-tensor $A_{abcd}$, the commutator convention induced by the stated curvature sign is \begin{align*} (\nabla_p\nabla_i-\nabla_i\nabla_p)A_{abcd} = - R_{pi a q}A_{qbcd} - R_{pi b q}A_{aqcd} - R_{pi c q}A_{abqd} - R_{pi d q}A_{abcq}. \end{align*} Applying this to $A_{abcd}=R_{abcd}$ gives \begin{align*} -\nabla_p\nabla_i R_{jpk\ell} = -\nabla_i\nabla_p R_{jpk\ell} + R_{pijq}R_{qpk\ell} + R_{pipq}R_{jqk\ell} + R_{pikq}R_{jpq\ell} + R_{pi\ell q}R_{jpkq}. \end{align*} and \begin{align*} -\nabla_p\nabla_j R_{pik\ell} = -\nabla_j\nabla_p R_{pik\ell} + R_{pjpq}R_{qik\ell} + R_{pj iq}R_{pqk\ell} + R_{pjkq}R_{piq\ell} + R_{pj\ell q}R_{pikq}. \end{align*} We derive the contracted Bianchi identities used here from the second Bianchi identity, so that the signs are fixed. Contract the second Bianchi identity in the first and fourth curvature slots and use the convention $\operatorname{Ric}_{ij}=R_{pij p}$. This gives the first contraction \begin{align*} \nabla_p R_{jpk\ell}=\nabla_\ell\operatorname{Ric}_{jk}-\nabla_k\operatorname{Ric}_{j\ell}. \end{align*} The same contraction with $j$ replaced by $i$ gives the second contraction \begin{align*} \nabla_p R_{pik\ell}=\nabla_k\operatorname{Ric}_{i\ell}-\nabla_\ell\operatorname{Ric}_{ik}. \end{align*} Therefore \begin{align*} -\nabla_i\nabla_p R_{jpk\ell} - \nabla_j\nabla_p R_{pik\ell} = -\nabla_i\nabla_\ell\operatorname{Ric}_{jk} + \nabla_i\nabla_k\operatorname{Ric}_{j\ell} -\nabla_j\nabla_k\operatorname{Ric}_{i\ell} + \nabla_j\nabla_\ell\operatorname{Ric}_{ik}. \end{align*} Moving these terms to the opposite side produces the second-Ricci combination displayed in the statement of the claim. It remains to simplify the curvature products. We use the curvature symmetries \begin{align*} R_{ijk\ell}=-R_{jik\ell}, \quad R_{ijk\ell}=-R_{ij\ell k}, \quad R_{ijk\ell}=R_{k\ell ij}, \end{align*} and the Ricci contraction $\operatorname{Ric}_{ij}=R_{pij p}$ with the present convention. The two Ricci contractions coming from the second index of the differentiated curvature are \begin{align*} -R_{pipq}R_{jqk\ell}=-\operatorname{Ric}_{ip}R_{pjk\ell}, \quad -R_{pjpq}R_{qik\ell}=-\operatorname{Ric}_{jp}R_{ipk\ell}. \end{align*} The terms in which the commutator acts on the first curvature index combine by pair symmetry and skew-symmetry as \begin{align*} -R_{pijq}R_{qpk\ell}-R_{pjiq}R_{pqk\ell}=-R_{ijpq}R_{k\ell pq}. \end{align*} The terms in which the commutator acts on the $k$ index give \begin{align*} -R_{pikq}R_{jpq\ell}-R_{pjkq}R_{piq\ell}=2R_{ipkq}R_{jp\ell q}-\operatorname{Ric}_{kp}R_{ijp\ell}. \end{align*} The terms in which the commutator acts on the $\ell$ index give \begin{align*} -R_{pi\ell q}R_{jpkq}-R_{pj\ell q}R_{pikq}=-2R_{ip\ell q}R_{jpkq}+\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} Each displayed equality is obtained only by renaming dummy indices and applying the three stated curvature symmetries; no unproved cancellation is being inserted. Hence the full commutator contribution is \begin{align*} -\left( R_{pijq}R_{qpk\ell} + R_{pipq}R_{jqk\ell} + R_{pikq}R_{jpq\ell} + R_{pi\ell q}R_{jpkq} \right) - \left( R_{pjpq}R_{qik\ell} + R_{pj iq}R_{pqk\ell} + R_{pjkq}R_{piq\ell} + R_{pj\ell q}R_{pikq} \right) = 2\left( R_{ipkq}R_{jp\ell q} - R_{ip\ell q}R_{jpkq} \right) - R_{ijpq}R_{k\ell pq} - \operatorname{Ric}_{ip}R_{pjk\ell} - \operatorname{Ric}_{jp}R_{ipk\ell} - \operatorname{Ric}_{kp}R_{ijp\ell} + \operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} Combining the previous two displays proves the claimed Bianchi commutation identity. [/proof][/step]

[guided]The delicate point is that the Laplacian identity is not a formal cancellation; it is a convention-dependent consequence of the second Bianchi identity and the curvature commutator. Fix $x_0 \in U$ and choose normal coordinates for $g(t)$ at $x_0$. Because all terms are tensorial, proving the identity at this point proves it on $U$. The second Bianchi identity says \begin{align*} \nabla_p R_{ijk\ell}+\nabla_i R_{jpk\ell}+\nabla_j R_{pik\ell}=0. \end{align*} Apply $\nabla_q$ and contract $q$ with $p$ using $g^{pq}$. Since $\nabla g=0$, this gives \begin{align*} \Delta R_{ijk\ell}=-\nabla_p\nabla_i R_{jpk\ell}-\nabla_p\nabla_j R_{pik\ell}. \end{align*} Now commute the first two covariant derivatives using the curvature commutator for a covariant $4$-tensor. This produces four curvature-product terms from the first summand and four from the second summand. The contracted Bianchi identities, derived by contracting the second Bianchi identity and using $\operatorname{Ric}_{ij}=R_{pij p}$, are \begin{align*} \nabla_p R_{jpk\ell}=\nabla_\ell\operatorname{Ric}_{jk}-\nabla_k\operatorname{Ric}_{j\ell} \end{align*} and \begin{align*} \nabla_p R_{pik\ell}=\nabla_k\operatorname{Ric}_{i\ell}-\nabla_\ell\operatorname{Ric}_{ik}. \end{align*} Substituting these two contractions gives exactly the second-Ricci derivative combination on the left side of the claimed identity. It remains to simplify the eight curvature products. Pairing the products according to the curvature index on which the commutator acts and using $R_{ijk\ell}=-R_{jik\ell}$, $R_{ijk\ell}=-R_{ij\ell k}$, and $R_{ijk\ell}=R_{k\ell ij}$ gives \begin{align*} -R_{pipq}R_{jqk\ell}-R_{pjpq}R_{qik\ell}=-\operatorname{Ric}_{ip}R_{pjk\ell}-\operatorname{Ric}_{jp}R_{ipk\ell}, \end{align*} \begin{align*} -R_{pijq}R_{qpk\ell}-R_{pjiq}R_{pqk\ell}=-R_{ijpq}R_{k\ell pq}, \end{align*} \begin{align*} -R_{pikq}R_{jpq\ell}-R_{pjkq}R_{piq\ell}=2R_{ipkq}R_{jp\ell q}-\operatorname{Ric}_{kp}R_{ijp\ell}, \end{align*} and \begin{align*} -R_{pi\ell q}R_{jpkq}-R_{pj\ell q}R_{pikq}=-2R_{ip\ell q}R_{jpkq}+\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} Adding these four reductions proves the displayed Bianchi commutation identity.[/guided]

[step:Substitute the Bianchi commutation identity into the variation formula]From the curvature variation formula, \begin{align*} \partial_t R_{ijk\ell} &= -\nabla_i\nabla_k\operatorname{Ric}_{j\ell} +\nabla_i\nabla_\ell\operatorname{Ric}_{jk} +\nabla_j\nabla_k\operatorname{Ric}_{i\ell} -\nabla_j\nabla_\ell\operatorname{Ric}_{ik} -R_{jikp}\operatorname{Ric}_{p\ell} -R_{ji\ell p}\operatorname{Ric}_{kp} -2\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} Substituting the Bianchi commutation identity gives \begin{align*} \partial_t R_{ijk\ell} = \Delta R_{ijk\ell} + 2\left( R_{ipkq}R_{jp\ell q} - R_{ip\ell q}R_{jpkq} \right) - R_{ijpq}R_{k\ell pq} - \operatorname{Ric}_{ip}R_{pjk\ell} - \operatorname{Ric}_{jp}R_{ipk\ell} - \operatorname{Ric}_{kp}R_{ijp\ell} + \operatorname{Ric}_{\ell p}R_{ijkp} - 2\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} Combining the final two terms yields \begin{align*} \partial_t R_{ijk\ell} = \Delta R_{ijk\ell} + 2\left( R_{ipkq}R_{jp\ell q} - R_{ip\ell q}R_{jpkq} \right) - R_{ijpq}R_{k\ell pq} - \operatorname{Ric}_{ip}R_{pjk\ell} - \operatorname{Ric}_{jp}R_{ipk\ell} - \operatorname{Ric}_{kp}R_{ijp\ell} - \operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} This is the stated evolution equation for the fully covariant Riemann curvature tensor.[/step]

[guided]The final step is bookkeeping, but the bookkeeping matters because the fully covariant tensor has one extra metric-lowering contribution. We begin from the corrected variation formula: \begin{align*} \partial_t R_{ijk\ell} &= -\nabla_i\nabla_k\operatorname{Ric}_{j\ell} +\nabla_i\nabla_\ell\operatorname{Ric}_{jk} +\nabla_j\nabla_k\operatorname{Ric}_{i\ell} -\nabla_j\nabla_\ell\operatorname{Ric}_{ik} -R_{jikp}\operatorname{Ric}_{p\ell} -R_{ji\ell p}\operatorname{Ric}_{kp} -2\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} The Bianchi commutation identity replaces every term except the final metric-lowering contribution by \begin{align*} \Delta R_{ijk\ell} + 2\left( R_{ipkq}R_{jp\ell q} - R_{ip\ell q}R_{jpkq} \right) - R_{ijpq}R_{k\ell pq} - \operatorname{Ric}_{ip}R_{pjk\ell} - \operatorname{Ric}_{jp}R_{ipk\ell} - \operatorname{Ric}_{kp}R_{ijp\ell} + \operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} Thus \begin{align*} \partial_t R_{ijk\ell} = \Delta R_{ijk\ell} + 2\left( R_{ipkq}R_{jp\ell q} - R_{ip\ell q}R_{jpkq} \right) - R_{ijpq}R_{k\ell pq} - \operatorname{Ric}_{ip}R_{pjk\ell} - \operatorname{Ric}_{jp}R_{ipk\ell} - \operatorname{Ric}_{kp}R_{ijp\ell} + \operatorname{Ric}_{\ell p}R_{ijkp} -2\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} The last two terms combine to $-\operatorname{Ric}_{\ell p}R_{ijkp}$, giving exactly \begin{align*} \partial_t R_{ijk\ell} = \Delta R_{ijk\ell} + 2\left(R_{ipkq}R_{jp\ell q}-R_{ip\ell q}R_{jpkq}\right) -R_{ijpq}R_{k\ell pq} -\operatorname{Ric}_{ip}R_{pjk\ell} -\operatorname{Ric}_{jp}R_{ipk\ell} -\operatorname{Ric}_{kp}R_{ijp\ell} -\operatorname{Ric}_{\ell p}R_{ijkp}. \end{align*} This is the claimed Hamilton evolution equation for the fully covariant Riemann curvature tensor.[/guided]