----- > [!proposition] Proposition. ([[characterization of lines of curvature (Olinde Rodrigues)]]) > A [[parameterized curve|curve]] $C$ in a [[differentiable Euclidean submanifold|regular surface]] $S$ is a [[line of curvature on a surface|line of curvature]] on $S$ if and only if for any parameterization $\alpha(t)$ of $C$ we have $N'(t)=-\lambda(t)\alpha'(t),$ > where $N(t)=N \circ \alpha(t)$ and $\lambda(t)$ is the [[principal curvature]] along $\frac{\alpha'(t)}{\|\alpha'(t)\|_{2}}$. Here, $N$ denotes the [[Gauss map]]. ^a5e53c > [!proof]- Proof. ([[characterization of lines of curvature (Olinde Rodrigues)]]) > Let $\alpha(t)$ be any [[parameterized curve|parameterization]] for $C$, then to say $C$ is a [[line of curvature on a surface|line of curvature]] on $S$ is to say that for all $t$, $\frac{\alpha'(t)}{\|\alpha'(t)\|_{2}}$ is a [[principal curvature|principal direction]], i.e., $\frac{\alpha'(t)}{\|\alpha'(t)\|}$ is an [[eigenvector]] of the [[differential of smooth maps between surfaces|differential]] $-dN_{p}$. Let $\lambda(t)$ be the associated [[eigenvalue]]. Then $dN_{p}(\alpha'(t))=-\lambda(t) \alpha'(t)$. The LHS is by definition $\frac{d}{dt}N(\alpha(t))$, i.e., $N'(t)$. We can check that $\lambda(t)$ is the [[principal curvature]] along $\frac{\alpha'(t)}{\|\alpha'(t)\|}$ thus: $\begin{align} \text{II}_{p}\left( \frac{\alpha'(t)}{\|\alpha'(t)\|}, \frac{\alpha'(t)}{\|\alpha'(t)\|} \right)= & -dN_{p}\left( \frac{\alpha'(t)}{\|\alpha'(t)\|} \right) \cdot \frac{\alpha'(t)}{\|\alpha'(t)\|}\\ = & \lambda(t) \frac{\alpha'(t)}{\|\alpha'(t)\|} \cdot \frac{\alpha'(t)}{\|\alpha'(t)\|}=\lambda(t). \end{align}$ ----- #### ---- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag