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...

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 is 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...