I'm trying to find a definition for Gauss-Kronecker curvature of submanifolds of $\Bbb R^n$, but I'm only finding it for hypersurfaces. I would like to know if someone knows any text which works in $\Bbb R^n$, but not kicking it and going all out for manifolds. (I'm self-studying this). I also have some ideas, which I would like to know if are in the right way.
Let $M^k \subset \Bbb R^n$ be a oriented submanifold. Let $\{{\bf e}_1, \cdots, {\bf e}_k\}$ and $\{{\bf N}_1, \cdots, {\bf N}_{n-k}\}$ be orthonormal bases for the tangent and normal spaces, respectively.
The second fundamental form at the point ${\bf p} \in M, {\rm II}: T_{\bf p}M \times T_{\bf p}M \to (T_{\bf p}M)^\perp$ can be defined by: $${\rm II}({\bf v},{\bf w}) = \sum_{i = 1}^{n-k} -\langle {\rm d}{{\bf N}_i({\bf v}),{\bf w}}\rangle {\bf N}_i.$$
Since $\{{\bf e}_i\}_{i=1}^k$ is an orthonormal basis, we define the trace of ${\rm II}$ by $${\rm tr}({\rm II}) = \sum_{i=1}^k {\rm II}({\bf e}_i,{\bf e}_i).$$
So we define the mean curvature vector by: $${\bf H} = \frac{1}{n}{\rm tr(II)}.$$
Here my problem begins. I'm trying to base myself in this part of Kühnel's Differential Geometry: Curves - Surfaces - Manifolds, in page $118$:
- I'm tempted to define the Gauss-Kronecker curvature as: $$K = \frac{\det_{\langle \cdot,\cdot \rangle}({\rm II})}{\det({\rm I})}.$$ But this notion of $\det_{\langle \cdot, \cdot \rangle}$ does not seems precise enough for me. This bothers me, because if the dimension of the submanifold is greater than $2$, what would be taking the "product" of three vectors? (the "product" of two vector is simply their inner product).
But what is in favor of this definition is the following: in $\Bbb R^3$, we have one shape operator $\mathcal{S}_{\bf N} = -{\rm d}{\bf N}$. Here, with codimension $n-k$, we will have $n-k$ shape operators related to the initial fixed base: $$\mathcal{S}_{{\bf N}_1} = -{\rm d}{\bf N}_1, \ldots ,\mathcal{S}_{{\bf N}_{n-k}} = -{\rm d}{\bf N}_{n-k}.$$ I made a few examples in $\Bbb R^4$, which seems to indicate that: $$\sum_{i=1}^{n-k}\det(-{\rm d}{\bf N}_i) = \frac{\det_{\langle \cdot, \cdot \rangle}({\rm II})}{\det({\rm I})}.$$ Since in $\Bbb R^3$ we have $K = \det(\mathcal{S}_{\bf N})$, it would be natural to define $K$ as this sum, at least for $n=4$.
A final remark: I do not want a definition which uses principal curvatures. I want a definition that I can use safely in Lorentz-Minkowski space $\mathbb{R}^n_1$ (there are timelike surfaces with no principal directions).
I'm sorry for the long post, but I really need some pointers here, and I wanted to share my efforts with you. Thanks for the attention.