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I'm currently reading about the 2nd fundamental form and sectional and Gauss curvature. The situation where all the sources I'm consulting agree is the following:

Let $S$ be a 2-dimensional embedded submanifold of $M := R^n$ (with the standard Euclidean metric) and assume there is a non-vanishing normal field $N$ on $S$, normalized to 1. Let $II$ be the 2nd fundamental form of $S$ relative to $N$, i.e. $II(v, w) := - \langle \nabla v N, w \rangle$, and $k_1, k_2$ be the principal curvatures (i.e. the eigenvalues of $II$). Then the Gauss curvature is defined as the determinant $\det II = k_1 k_2$ and agrees with the sectional curvature $K = \langle R(X, Y)Y, X \rangle$ of $S$ (where $X, Y$ is any local orthonormal basis field of $TS$). In particular, the Gauss curvature is intrinsic to $S$ and does not depend on its embedding.

Now consider the case of higher-dimensional $S$ and/or a general ambient (Riemannian) manifold $M$. In this case, most sources I've consulted don't define Gauss curvature anymore but exclusively use sectional curvature. Others (like Jeffrey M. Lee's "Manifolds & Differential Geometry") still define Gauss curvature in the case of higher-dimensional $S$ and $M = R^n$ (but not for general $M$), namely as the determinant of $II$. Even others (like Wikipedia) take the products $k_i k_j$ of principal curvatures to be something like a "Gauss curvature in the i-j principal direction" (see below).

My confusion now comes from the following statement on Wikipedia in this context:

Similarly, if $S$ is a hypersurface in a Riemannian manifold $M$, then the principal curvatures are the eigenvalues of its second fundamental form. If $k_1, …, k_n$ are the $n$ principal curvatures at a point $p \in M$ and $X_1,…, X_n$ are corresponding orthonormal eigenvectors (principal directions), then the sectional curvature of $S$ at $p$ is given by $K(X_i, X_j) = k_i k_j$.

(Note that I renamed the manifolds to match my notation from above.)

How can the statement from Wikipedia be true, though, if the ambient manifold $M$ is curved? In this case, the Gauss equation$^\dagger$

$$ R^S(X, Y, Z, W) = R^M(X, Y, Z, W) - II(X, Z) II(Y, W) + II(X, W) II(Y, Z) \\ \forall p \in S \text{ and } X, Y, Z, W \in T_pS $$

clearly gives an additional term arising from the curvature of $M$, spoiling the equality between sectional curvature and the products $k_i k_j$ (i.e. the determinants of the 2-dimensional minors of $II$, viewed as a matrix $II_ij$ w.r.t. the principal directions). Since the principal curvatures $k_i$ are usually interpreted as the extrema of the normal curvature – the normal curvature being the extrinsic curvature of the curve obtained from intersecting $S$ with the corresponding normal plane – I suspect that this correctional term $R^M(X, Y, Z, W)$ can be interpreted in the sense that the normal planes cannot be actual planes anymore but are curved themselves due to living in the curved manifold $M$. So the entire interpretation of the eigenvalues of $II$ somewhat breaks down.

My questions are:

  1. Is the statement on Wikipedia indeed wrong?

and, assuming that the answer is "yes":

  1. Is my interpretation of the correctional term in the Gauss equation correct?
  2. What is the / Is there a good interpretation of the eigenvalues of $II$ and the determinant $\det II$ in the situation of a curved ambient manifold and/or a higher-dimensional $S$?

and, if you allow me a somewhat subjective question:

  1. Does it even make sense to define a "Gauss curvature" in the general setting (when we're not in the situation of flat $M$ and $dim S = 2$), considering that we already define sectional curvature and this seems to be all we need (since it is the suitable generalization of intrinsic curvature of 2-dimensional submanifolds)?

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$^\dagger$ Note that I'm restricting the Gauss equation to the case where $S$ has codimension 1 and the normal field $N$ is unique up to a sign (provided it exists). Otherwise, the products of terms $II(X, Y)$ obviously have to be replaced with inner products of terms $(\nabla_X Y)^\perp$, the normal part of the covariant derivative on $M$.

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