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.
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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: $${\rm II}({\bf v},{\bf w}) = \sum_{i = 1}^{n-k} -\langle {\rm d}{{\bf N}_i({\bf v}),{\bf w}}\rangle {\bf N}_i.$$
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Since $\{{\bf e}_i\}_{i=1}^k$ is an orthonormal basis, we define the trace of ${\rm II}$ by $${\rm tr}({\rm II}) = \sum_{i=1}^k {\rm II}({\bf e}_i,{\bf e}_i).$$
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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.
Best Answer
One possible reason for you to be having trouble finding the 'right' definition of Gauss-Kronecker curvature is that you haven't really told us what properties you want this curvature to have. I gather that you want it to be a scalar that is a function of the second fundamental form somehow and that, in the hypersurface case, it should be the classical Gauss-Kronecker curvature, which, in the hypersurface case, is defined as the ratio of the $\mathbf{N}_1$-pullback of the volume form on $S^{n-1}$ (where $\mathbf{N}_1:M\to S^{n-1}$ is the (oriented) Gauss map) to the induced volume form on $M$.
The natural way to generalize this is to generalize the Gauss map. In other words, for an oriented manifold and frame field as you have defined it above, consider the mapping $$ ^\ast\gamma = \mathbf{N}_1\wedge \mathbf{N}_2\wedge\cdots \wedge\mathbf{N}_{n-k}: M\to \mathrm{Gr}_{n-k}(\mathbb{R}^n) $$ or, what, by duality, is essentially the same thing, the generalized Gauss mapping $$ \gamma = \mathbf{e}_1\wedge \mathbf{e}_2\wedge\cdots \wedge\mathbf{e}_{k}: M\to \mathrm{Gr}_{k}(\mathbb{R}^n), $$ where $\mathrm{Gr}_{p}(\mathbb{R}^n)$ means the Grassmannian of oriented $p$-planes in $\mathbb{R}^n$, which is a homogeneous space of $\mathrm{SO}(n)$ of dimension $p(n{-}p)$ and which carries a natural $\mathrm{SO}(n)$-invariant metric, $h$.
Then one natural definition of a scalar $S$ is to take the ratio of the volume form of $\gamma^*h$ to the induced volume form on $M$. Up to a sign, this reduces to the Gauss-Kronecker curvature when $k=n{-}1$ (i.e., the hypersurface case). The formula in terms of the coefficients of the second fundamental form is then $$ S = \sqrt{\det(H_{j l})} \qquad\text{where}\qquad H_{jl} = \sum_{\alpha, i} h_{\alpha i j}h_{\alpha i l} $$ where the index $\alpha$ satisfies $1\le \alpha\le n{-}k$ and the latin indices satisfy $1\le i,j\le k$.
Another natural scalar that is well-defined when $k=2$ or $k=n{-}2$ and $n$ is even is to note that $\mathrm{Gr}_{2}(\mathbb{R}^n)$ and $\mathrm{Gr}_{n-2}(\mathbb{R}^n)$ both carry a canonical $\mathrm{SO}(n)$-invariant $2$-form, $\Omega$. When $k=2$, you could just define a scalar $W$ to be the ratio of $\gamma^*\Omega$ to the induced area form on $M^2$. The formula is $$ W = \sum_{\alpha} h_{\alpha11}h_{\alpha22}-{h_{\alpha12}}^2 $$ This agrees with the Gauss-Kronecker curvature when $n=3$ and generalizes it when $n>3$. When $n=2m$ and $k=n{-}2=2m{-}2$, you could take $L$ to be the scalar that is the ratio of $(\gamma^*\Omega)^{m-1}/((m{-}1)!)$ to the volume form on $M$. Of course, $W$ and $L$ are the same when $(k,n) = (2,4)$.
Of course, $S$, $W$, and $L$ generally have different properties, even when they are all defined. The good thing is that these things don't depend on a positive definite metric. They'll work for any submanifold on which the first fundamental form is nondegenerate.