[proofplan] We use the coordinate form of the Ricci-DeTurck equation, whose principal part is the strictly parabolic operator $g^{ij}\partial_{x_i}\partial_{x_j}$ acting on the metric components. The lower-order terms define a smooth quasilinear expression on the open cone of positive-definite symmetric tensors. A standard short-time existence theorem for quasilinear strictly parabolic systems on vector bundles over compact manifolds then gives a unique solution in parabolic Hölder spaces. Positivity persists for small time, and parabolic bootstrapping upgrades the solution to smoothness. [/proofplan]

[step:Write the Ricci-DeTurck equation as a quasilinear parabolic system on $S^2T^*M$]Let $E := S^2T^*M$ denote the smooth vector bundle of symmetric covariant $2$-tensors on $M$. Let \begin{align*} \mathcal{P} \subset E \end{align*} denote the open subbundle whose fiber $\mathcal{P}_p \subset S^2T_p^*M$ consists of positive-definite symmetric bilinear forms on $T_pM$. In a coordinate chart $(U,\varphi)$ with coordinates $(x_1,\dots,x_n)$, write a time-dependent symmetric tensor as \begin{align*} g(t)|_U = \sum_{a,b=1}^n g_{ab}(\cdot,t)\, dx_a \otimes dx_b. \end{align*} The Ricci-DeTurck equation with background metric $\bar{g}=g_0$ has the local form \begin{align*} \partial_t g_{ab} = g^{ij}\partial_{x_i}\partial_{x_j}g_{ab} + Q_{ab}(x,g,\partial_x g), \end{align*} where each $Q_{ab}$ is a smooth function of the coordinate point $x$, the matrix entries of $g$, the inverse matrix entries of $g$, the first derivatives of $g$, and the fixed smooth coefficients of $\bar{g}$ and their derivatives. This is the coordinate expression supplied by the preceding Ricci-DeTurck parabolicity computation, citing a result not yet in the wiki: [Strict Parabolicity of the Ricci-DeTurck Flow](/theorems/5970). For every $p \in M$, every positive-definite tensor $h \in \mathcal{P}_p$, and every nonzero covector $\xi \in T_p^*M$, the principal symbol of the operator acting on $S^2T_p^*M$ is \begin{align*} \sigma_p(\xi)(v) = h^{ij}\xi_i\xi_j\, v, \qquad v \in S^2T_p^*M. \end{align*} Since $h$ is positive definite, $h^{ij}\xi_i\xi_j > 0$ for $\xi \ne 0$. Hence the system is strictly parabolic whenever $g(t)$ remains in $\mathcal{P}$.[/step]

[guided]The point of the DeTurck modification is that it changes the Ricci flow equation into a genuinely parabolic system for the metric components. We work on the vector bundle \begin{align*} E := S^2T^*M, \end{align*} whose sections are smooth symmetric covariant $2$-tensors. The Riemannian metrics are not all of $E$; they form the open subbundle \begin{align*} \mathcal{P} \subset E \end{align*} whose fiber consists of positive-definite symmetric bilinear forms. Choose a coordinate chart $(U,\varphi)$ with coordinates $(x_1,\dots,x_n)$. On this chart, a time-dependent symmetric tensor has the component expression \begin{align*} g(t)|_U = \sum_{a,b=1}^n g_{ab}(\cdot,t)\, dx_a \otimes dx_b. \end{align*} The coordinate form of the Ricci-DeTurck equation is \begin{align*} \partial_t g_{ab} = g^{ij}\partial_{x_i}\partial_{x_j}g_{ab} + Q_{ab}(x,g,\partial_x g). \end{align*} Here $g^{ij}$ denotes the inverse matrix of $g_{ij}$, and $Q_{ab}$ denotes the lower-order expression depending smoothly on $x$, the entries of $g$, the entries of $g^{-1}$, the first derivatives of $g$, and the fixed smooth background metric $\bar{g}=g_0$. This coordinate reduction is precisely the parabolicity computation for the Ricci-DeTurck equation, citing a result not yet in the wiki: Strict Parabolicity of the Ricci-DeTurck Flow. Why does this imply strict parabolicity? The second-order part is the scalar operator $g^{ij}\partial_{x_i}\partial_{x_j}$ applied separately to each component $g_{ab}$. Thus, at a point $p \in M$, for a positive-definite tensor $h \in \mathcal{P}_p$ and a nonzero covector $\xi \in T_p^*M$, the principal symbol on the fiber $S^2T_p^*M$ is \begin{align*} \sigma_p(\xi)(v) = h^{ij}\xi_i\xi_j\, v, \qquad v \in S^2T_p^*M. \end{align*} Since $h$ is positive definite, the inverse tensor $h^{-1}$ is positive definite on covectors, so \begin{align*} h^{ij}\xi_i\xi_j > 0 \end{align*} whenever $\xi \ne 0$. Therefore the principal symbol is a positive scalar multiple of the identity on $S^2T_p^*M$. This is the strict parabolicity required by quasilinear parabolic existence theory.[/guided]

[step:Apply short-time existence for strictly parabolic quasilinear systems] We apply the standard short-time existence theorem for smooth quasilinear strictly parabolic systems on vector bundles over compact manifolds, citing a result not yet in the wiki: Short-Time Existence for Quasilinear Strictly Parabolic Systems. The theorem applies because $M$ is compact without boundary, the bundle $E=S^2T^*M$ is smooth and finite-rank, the initial section $g_0 \in \Gamma(E)$ is smooth, the nonlinearity \begin{align*} (g,\partial_x g,\partial_x^2 g) \mapsto g^{ij}\partial_{x_i}\partial_{x_j}g_{ab}+Q_{ab}(x,g,\partial_x g) \end{align*} is smooth on the [open set](/page/Open%20Set) $\mathcal{P}$ of positive-definite tensors, and the principal symbol is strictly positive by the previous step. Hence there exist a time $T_1>0$ and a unique solution \begin{align*} g_*: M \times [0,T_1] &\to E \end{align*} in a parabolic Hölder class, for example $C^{2+\alpha,1+\alpha/2}$ for some fixed $\alpha \in (0,1)$, satisfying the Ricci-DeTurck equation and the initial condition $g_*(0)=g_0$. [/step]

[step:Shrink the time interval so the solution remains a Riemannian metric] Because $g_0$ is a Riemannian metric and $M$ is compact, there exists a constant $\lambda_0>0$ such that \begin{align*} g_0(v,v) \geq \lambda_0 |v|_{g_0}^2 \end{align*} for every $p \in M$ and every $v \in T_pM$. In the norm induced by $g_0$, this inequality holds with $\lambda_0=1$. The section $g_* \in C^{2+\alpha,1+\alpha/2}$ is continuous on the [compact space](/page/Compact%20Space)-time set $M \times [0,T_1]$. Therefore there exists $T \in (0,T_1]$ such that \begin{align*} |g_*(p,t)-g_0(p)|_{g_0} < \frac{1}{2} \end{align*} for every $p \in M$ and every $t \in [0,T]$, where $|\cdot|_{g_0}$ is the fiber norm on $S^2T^*M$ induced by $g_0$. It follows that for every $v \in T_pM$, \begin{align*} g_*(p,t)(v,v) \geq g_0(p)(v,v)-|g_*(p,t)-g_0(p)|_{g_0}|v|_{g_0}^2 > \frac{1}{2}|v|_{g_0}^2. \end{align*} Thus $g_*(t)$ is positive definite for every $t \in [0,T]$. Define $\hat{g}$ as the section-valued map \begin{align*} \hat{g}: M \times [0,T] &\to S^2T^*M. \end{align*} For each $(p,t) \in M \times [0,T]$, set $\hat{g}(p,t) := g_*(p,t)$. Then $\hat{g}(t)$ is a Riemannian metric for every $t \in [0,T]$. [/step]

[step:Bootstrap the parabolic solution to smoothness] The coefficients of the Ricci-DeTurck equation are smooth functions of $x$, $\hat{g}$, $\hat{g}^{-1}$, and the first derivatives of $\hat{g}$, and the solution remains in the positive-definite cone. Since the equation is strictly parabolic on $M \times [0,T]$, interior and initial-time parabolic regularity for smooth quasilinear systems apply iteratively, citing a result not yet in the wiki: Parabolic Schauder Bootstrapping for Smooth Quasilinear Systems. Starting from $\hat{g} \in C^{2+\alpha,1+\alpha/2}$, the equation expresses $\partial_t\hat{g}$ and the spatial second derivatives of $\hat{g}$ through smooth functions of lower-order quantities. Applying the [regularity theorem](/theorems/2750) repeatedly gives \begin{align*} \hat{g} \in C^\infty(M \times [0,T]). \end{align*} The compatibility at $t=0$ is automatic because the initial metric $g_0$ is smooth and the right-hand side of the equation is a smooth differential expression in $g_0$ and the fixed background metric $\bar{g}=g_0$. [/step]

[step:Use parabolic uniqueness to obtain uniqueness of the Ricci-DeTurck solution] Let \begin{align*} g_1,g_2: M \times [0,T] &\to S^2T^*M \end{align*} be two smooth Riemannian metric solutions of the Ricci-DeTurck equation with $g_1(0)=g_2(0)=g_0$. Since each solution remains positive definite, both solve the same strictly parabolic quasilinear system on the open set $\mathcal{P} \subset S^2T^*M$. By the uniqueness part of the short-time existence theorem for quasilinear strictly parabolic systems, applied on the compact manifold $M$, the two solutions agree on their common time interval: \begin{align*} g_1(p,t)=g_2(p,t) \end{align*} for every $p \in M$ and every $t \in [0,T]$. Therefore the smooth solution $\hat{g}$ constructed above is unique. This proves short-time existence and uniqueness for the Ricci-DeTurck flow with background metric $\bar{g}=g_0$. [/step]