Several authors (also of standard RiemGeo books) write that the sectional curvature of a plane $\pi$ contained in $T_pM$, where $(M,g)$ is a Riemannian manifold of any dimension, is the "Gaussian curvature" in $p$ of the surface $S$ generated by the geodesics starting at $p$, with tangent velocity belonging to $\pi$. Then,unfortunately and surprisingly, they define the Gaussian curvature for surfaces embedded in $\mathbb R^3$. I do not really have an idea why, since the situation is completely different. You actually need a definition of Gaussian curvature for abstract surfaces or at least for surfaces embedded in a generic Riemannian manifold (at least $\mathbb R^n$, with $n\geq 4$, by Nash embedding Thm that would be sufficient) with any codimension, not only for embedded surfaces in $\mathbb R^3$, since not every abstract surface can be isometrically embedded in $\mathbb R^3$. Hence, also mentioning the "Theorema egregium" has apparently nothing to do with that, since it deals with surfaces in $\mathbb R^3$, and this is completely not the case . Moreover they usually say that this was the way (what way? It is not defined!) Riemann generalizes to abstract manifold the concept of curvature... I never saw the original papers of Riemann, so I cannot say if this i true or not, but something is not quite clear here. Hence, here are my (first) questions:

• What is the definition of Gaussian curvature for an abstract surface? At least for isometrically embedded surfaces in $\mathbb R^n$, with $n\geq 4$? - Definition almost impossible to be found clear in these books (up to my knowledge... if I am wrong, my mistake) and around the web...

• How Riemann really defined sectional curvatures? Does someone really know?

• Why people mention the "Theorema egregium", that deals with a completely different situation? Clearly not this one...