# Gaussian curvature versus sectional curvature

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?

• To add to Thomas' good answer: The books I've used for differential geometry use the term normal curvature for the extrinsic curvature of a normal section of a surface in $\mathbf{R}^{3}$. Thus, the principal curvatures are the extremes of the normal curvatures. (The term sectional curvature is well-established for a completely different concept: The intrinsic curvature of a $2$-plane (i.e., the curvature of the image of a small piece of tangent plane under the exponential map) in a Riemannian manifold of dimension at least $2$.) Mar 27, 2016 at 11:29
• The term "sectional curvature" refers to the ordinary Gaussian curvature of a totally geodesic (2-dimensional) surface through a point and tangent to a given 2-plane, in a higher-dimensional manifold. (It is not used when the manifold is itself a surface, since there is only one tangent 2-plane for each point.) Apr 4, 2023 at 19:41
• @Dan Asimov, I would be careful, in Lee's introduction to Riemannian manifolds (1st and 2nd edition), the term "sectional curvature" is defined for $n$-dimensionnal Riemannian manifolds, where $n\geq 2$.
– lou
Mar 16 at 4:16
• Sure, it's defined, but Lee or no Lee, people talking or writing about a 2-manifold don't use the term "sectional" curvature in practice, because there is only one at each point. Mar 16 at 8:46

Some lines before that statement they say what they mean by a 'normal section' and denote the curvature of the resulting line by 'normal curvature' of that normal section. What is true is that, in the hyperbolic case, there are two directions for which the corresponding normal curvatures are $=0$ (because one principal curvature is negative, one is positive, and the normal section varies continuously if you rotate the plane orthogonal to the surface around the normal to the surface). The use of the name 'sectional curvature' is incorrect here, but very likey deemed natural to the author since the curvature he has in mind is derived from intersecting the surface with a normal plane.