I was studying https://en.wikipedia.org/wiki/Gaussian_curvature (exact version https://en.wikipedia.org/w/index.php?title=Gaussian_curvature&oldid=709607678 ) and there it says: (bold added)
The sign of the Gaussian curvature can be used to characterise the surface.
- If both principal curvatures are the same sign: $κ_1κ_2 > 0 $, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign.
- If the principal curvatures have different signs: $κ_1κ_2 < 0 $, then the Gaussian curvature is negative and the surface is said to have a hyperbolic point. At such points the surface will be saddle shaped. For two directions the sectional curvatures will be zero giving the asymptotic directions.
- If one of the principal curvatures is zero: $κ_1κ_2 = 0$, the Gaussian curvature is zero and the surface is said to have a parabolic point.
I don't undterstand the For two directions the sectional curvatures will be zero giving the asymptotic directions.
I understand that for surfaces with a negative Gaussian curvature there are two directions where that are in the tangent plane but what does that has to do with the sectional curvature? ( I thought that in 2 dimensional manifolds the sectional and Gausian curvature are the same curvature)
Also is there someting important relating to these two directions?