Continuous deformation from 3-dim Euclidean space $y^\mu=x^\mu-u^\mu(\boldsymbol{x})$ implys trivial Riemann curvature tensor. It is clear to see that a single-valued continuous displacement field $u^\mu$ is promised. \begin{equation} R^\lambda_{\ \ \sigma\mu\nu}=0 \Leftrightarrow [\nabla_\mu\nabla_\nu-\nabla_\nu\nabla_\mu]u^\lambda=0 \end{equation} I want to verify it, but when I plug the metric in deformed medium $g_{\mu\nu}=\delta_{\mu\nu}+2\epsilon_{\mu\nu}$, where $\epsilon_{\mu\nu}=\partial_\mu u_\nu+\partial_\nu u_\mu$ is the infinitesimal strain tensor, back in to the Riemann curvature tensor \begin{equation} R_{\lambda \sigma \mu \nu}= \partial_{\mu} \Gamma_{\lambda\nu \sigma}-\partial_{\nu} \Gamma_{\lambda \mu \sigma}+g_{\rho\gamma}(\Gamma_{\ \ \lambda\nu}^{\rho}\Gamma_{\ \ \sigma\mu}^{\gamma} - \Gamma_{\ \ \lambda\mu }^{\rho}\Gamma_{\ \ \sigma\nu}^{\gamma}) \end{equation} I got some weird constriants from the products of Christoffel symbols above: \begin{equation} \partial_{\nu}\partial_{\lambda}u_\alpha \partial_{\mu}\partial_{\sigma}u_\beta - \partial_{\mu}\partial_{\lambda}u_\alpha \partial_{\nu}\partial_{\sigma}u_\beta=0 \end{equation} for $\Gamma_{\lambda\mu\nu}=\frac{1}{2}(\partial_{\mu}\partial_{\nu}u_\lambda+\partial_{\nu}\partial_{\mu}u_\lambda)$. I summerize my problems as below

  1. Dose it mean an exart constraint on the displacement field $\boldsymbol{u}$? Or I just made some stupid mistakes somewhere?
  2. If directly plug $g_{\mu\nu}=\delta_{\mu\nu}+2\epsilon_{\mu\nu}$ into curvature tensor, I should get some constraints on $\epsilon_{\mu\nu}$ similarly, that is compatibility conditions(related to Saint-Venant tensor). It looks like(from Chapter 4.3 in Hashiguchi(2012)): \begin{equation} \left(\frac{\partial^{2} e_{i l}}{\partial \theta^{j} \partial \theta^{k}}+\frac{\partial^{2} e_{j k}}{\partial \theta^{i} \partial \theta^{l}}-\frac{\partial^{2} e_{i k}}{\partial \theta^{j} \partial \theta^{l}}-\frac{\partial^{2} e_{j l}}{\partial \theta^{i} \partial \theta^{k}}\right)=0 \end{equation}

It comes from the derivative terms of Christoffel symbols in curvature tensor. Due to the same difficulty, I don't know how to eliminate the product of Christoffel symbols. Any comment would be appreciated!


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