Curve and Surface Curvature: Formulas, Intrinsic Geometry, and Generalizations

All flashcards

How does orientation affect signed curvature?

Reversing the orientation reverses the sign of signed curvature, but not its magnitude.

What does the shape operator encode?

It describes how the surface normal changes; its eigenvalues are principal curvatures and its eigenvectors are principal directions.

How are mean curvature and principal curvatures related?

Mean curvature is half their sum: $H=(k_1+k_2)/2$.

What is holonomy in relation to curvature?

It describes how parallel transport around a closed loop can return a vector with a changed direction.

Quiz questions

1. For a regular space curve $\gamma(t)$, which formula gives its curvature?Answer: $\kappa=\frac{\|\gamma'(t)\times\gamma''(t)\|}{\|\gamma'(t)\|^3}$
2. What happens to ordinary curvature when a regular curve is reparametrized with a different speed?Answer: It remains unchanged at each corresponding point on the curve.
3. What characterizes a geodesic on a smooth surface?Answer: Its geodesic curvature is zero.
4. At a point on a smooth surface, what are the principal curvatures?Answer: The maximum and minimum normal curvatures over tangent directions.
5. For an arc-length-parametrized curve, how is the curvature vector $K$ related to the unit tangent $T$?Answer: $K=T'$