Undergrad What is the difference between Gaussian and sectional curvature?

[PhysicsRock]

TL;DR

See title

In a homework problem, I had to derive the relationship $R_{\mu\nu} = \pm K g_{\mu\nu}$ on a surface, i.e. a $2$-dimensional submanifold of $\mathbb{R}^3$. Here, $K$ is the Gaussian curvature. I think I managed to do that, but from my derivation I don't see why this result is restricted to the mentioned special case. I believe it has to do with the fact that I used Gaussian curvature explicitly, not sectional curvature as one would in the general case. However, I don't really understand the difference between the two. I can imagine that Gaussian curvature is a global property, which would allow for us to say that $R_{\mu\nu} = K g_{\mu\nu}$ is true everywhere on a such a manifold, whereas in the general case this might not be true in every chart, we still might find some where this holds.

Thank you in advance for any help.

A quick disclaimer at the end. Although this question arose from a homework problem, it's about my understanding in general and not about said problem specifically. Therefore, I thought it would be asked here best, not in the homework forums.

 

[andrewkirk]

I think that for a 2D manifold M, the sectional curvature (SC) at a point P will always equal the Gaussian curvature (GC) at P, because the SC is the GC at P of the submanifold S covered by applying the exponential map to the vectors in the tangent plane T_PM radiating away from P.

Except perhaps in pathological cases, the submanifold S will include an open neighbourhood of P in M, so the GC at P in S (which is the sectional curvature) will be the same as the GC at P in M.

Where M has dimension m > 2, that result will no longer apply, as S will be a proper submanifold of M on any neighbourhood of P. The relationship of SC to GC will depend on the choice of two tangent vectors to define the tangent plane at P: the vectors u and v in this article about sectional curvature.

In general neither SC nor GC would be a global property.