Let $M^n\subset \mathbb{R}^{n+1}$ be an embedded hypersurface and let $g$ be the metric induced on $M$ by the flat metric on $\mathbb{R}^{n+1}$.
Q: What is a formula for the Weyl tensor $W$ of $g$ in terms of the second fundamental form $A$ of $M$? More precisely, what is $|W|^2$?
For example, the scalar curvature $S$ of $g$ can be written as
$$
S=H^2-|A|^2=\mathrm{trace}(A)^2-\mathrm{trace}(A^2)=\sum_{1\leq i_1<i_2\leq n}\lambda_{i_1}\lambda_{i_2},
$$
where $H$ is the mean curvature (trace of $A$) and $\lambda_1,\dots,\lambda_n\in \mathbb{R}$ are the principal curvatures of $M$, i.e. the eigenvalues of $A$. Note that $S$ is thus an elementary symmetric polynomial in $\lambda_1,\dots,\lambda_n$.
Is $|W|^2$ also given by such a polynomial? I guess this is probably not true as $|W|^2$ is zero for any 2-manifold (all surfaces are locally conformally flat).
Best Answer
You can observe that by Gauss Equation and Codazzi-Mainardi equation you can write the Riemann tensor and the Ricci tensor in function of the second fundamental form of your hypersurface that is in $\mathbb{R}^{n+1}$, so the Riemann and Ricci tensor of the ambient are zero. The Weyl tensor is defined in function of the induced metric, the Riemann tensor and the Ricci tensor of the hypersurface so it is true, it is possible write the Weyl tensor in function of the second fundamental form.
We know that Gauss equation of an Hypersurface $M\subseteq N$ is
$R^N(X,Y,Z,W)=R^M(X,Y,Z,W)+h(X,W)h(Y,Z)-h(X,Z)h(Y,W)$
where $h$ is the second fundamental form of $M$.
In local coordinates we have that
$R^N_{ijkl}=R^M_{ijkl}+h_{il}h_{jk}-h_{ik}h_{jl}$
In our case $R^N=0$ so
$R^M_{ijkl}= h_{ik}h_{jl}-h_{il}h_{jk}$
If we trace the equation we can write also Ricci Tensor in function of the second fundamental form:
$R^M_{jl}=Hh_{jl}-g^{ik} h_{il}h_{jk}$
where $H$ is the mean curvature of $M$.
If we trace twice Gauss equation we get a relationship between the scalar curvature $R$ of $M$ and the second fundamental form:
$R=H^2-|h|^2$
We consider the (0,4)-Weyl tensor of $M$
$W=R^M-\frac{1}{n-2}(Ric-\frac{R}{n}g)\wedge g-\frac{R}{2n(n-1)}g\wedge g$
where $\wedge$ is the Kulkarni-Nomizou product.
In local coordinate we have that
$W_{ijkl}=R^M_{ijkl}+\frac{1}{n-2}(R^M_{il}g_{jk}-R^M_{ik}g_{il}+R^M_{jk}g_{il}-R^M_{jl}g_{ik})+$
$+\frac{R}{(n-1)(n-2)}(g_{ik}g_{jl}-g_{il}g_{jk})$
So you have all elements to write Weyl tensor in function of the second fundamental form of $M$ (and in function of the metric $g$).
You can observe also that the (0,2)-Tensor $h$ is symmetric so there exists a local orthonormal frame $\{Z_1,\dots, Z_n\}$ such that the matrix $A$ associated with $h$ with respect that frame is $diag(\lambda_1,\dots, \lambda_n)$ and th matrix $G$ associated with the metric $g$ with respect that frame will be the identity $I$. In this case you can write all our formula in an easier way than before.
If it is not clear I can explain more precisely.