When nothing deforms

For an admissible deformation gradient $\mathbf{F}$ at a point ($\det \mathbf{F}>0$), the following are equivalent:

1. every fibre keeps its length: ${\nu }_m=1$ for every unit $\mathbf{m}$;
2. $\mathbf{C}={\mathbf{F}}^T\mathbf{F}=\mathbf{I}$;
3. $\mathbf{F}$ is a rotation: $\mathbf{F}=\mathbf{R}$, with ${\mathbf{R}}^T\mathbf{R}=\mathbf{I}$, $\det \mathbf{R}=1$;
4. the right stretch tensor is the identity: $\mathbf{U}=\mathbf{I}$ (and then $\mathbf{V}=\mathbf{I}$);
5. every principal stretch is 1;
6. the Green–Lagrange strain vanishes: $\mathbf{E}=0$;
7. every fibre keeps its length and every right angle stays right: ${\nu }_m=1$ and ${\gamma }_{mn}=0$ for all orthogonal $\mathbf{m},\mathbf{n}$.