# Definition of Gauss curvature in the general setting

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 $$M$$.