A question about second fundamental form of Riemannian isometric embedding I moved the question from mathoverflow to here.
I have got a question unsolved for some time. I do not know whether it is trivial or not:
The metric at point p is second-order flat, i.e. $d_p \phi(-,v) = 0$ and $d_p^2 \phi(-, v) = 0$. $d_p$ is differential of a function at p and $d_p^2$ is second differentiation.
Consider an isometric embedding $f: (U, \phi) \rightarrow (\mathbb{R}^4, \Phi)$, here $\phi$ is norm of 3-dim manifold $U$ and $\Phi$ is norm of Euclidean space $\mathbb{R}^4$. We define 2nd fundamental form as symmetric bilinear form $S(v, w) = <d_p^2 f(v,w), N>$. Here N is normal vector at point $f(p)$ of manifold $f(U)$. Here metric at point p is second-order flat.
Problem:

*

*Is it true that: there exists $\tau \in \mathbb{R}^4 \setminus Im(d_p f)$ such that ${d_p}^2 f(v, w) = S(v, w) \tau$ for all $v, w \in \mathbb{R}^3$ ?


*Above question has an equivalent form: if 2nd fundamental form $S(v, w) = 0$ for linearly independent $v, w$, then $d_p^2 f(v, w) = 0$. Is it possible that $d_p^2 f(v, w)$ is nonzero tangent vector in the tangent space for some $v, w$ at a point of flat metric in 3-dim manifold? In this case this question can be proved incorrect.
Remark: The existence in problems 1, 2 can be proved when we embed 2-dim manifold into 3-dim Euclidean space, i.e. $f: (U, \phi) \rightarrow (\mathbb{R}^3, \Phi)$ and $U$ is 2-dim manifold.
More about notations:

*

*For notations $d_p^2 f(v, w)$, We first construct a chart on $r: R^3 \rightarrow U$ and this induces a map between charts $\tilde{f}: R^3 \rightarrow R^4$, so we just take second derivative of this map.


*In Do Carmo's differential geometry, if $r$ is map of embedding of (n-1)-dim manifold M into Euclidean space $\mathbb{R}^n$ (suppose such embedding exists), we use $r_{ij} = \Gamma_{ij}^k r_k + L_{ij} N $ and we say that $\Gamma_{ij}^k$ is Christopher symbol and $N$ is normal vector. So I think that projection of second derivative  $r_{ij}$ onto the tangent space is just Levi-Civita connection$\nabla_{r_i} r_j$. So here we can use $\nabla_{df(v)} {df(w)}$ at $p$ to define projection of $d_p^2 f(v, w)$ onto tangent space.
 A: $\newcommand\R{\mathbb{R}}$ Normally, I like using coordinate-free notation, but I find yours difficult to understand. So I'll write everything with respect to local coordinates. Let $U \subset \R^{n-1}$ be open and $\phi = \phi_{ij}\,dx^i\,dx^j$ be a Riemannian metric on $U$. Let $\Phi$ be the standard flat Riemannian metric on $\R^n$, i.e., the standard dot product.
If a map $f: (U, \phi) \rightarrow (\R^n,\Phi)$ is isometric, then
$$ \partial_if\cdot \partial_j f = \phi_{ij}. $$
As you say, the Hessian of $f$ can be decomposed into its tangential and normal components. If $N$ is a unit normal map (i.e., Gauss map), then, since $(\partial_1f, \dots, \partial_{n-1}f)$ is a basis of $T_p \subset \R^n$, there exist coefficients $\Gamma^k_{ij}, S_{ij}$ such that
$$ \partial^2_{ij}f = \Gamma^k_{ij}\partial_kf + S_{ij}N. $$
You can confirm that indeed the $\Gamma^k_{ij}$ are the Christoffel symbols for the Levi-Civita connection of $\phi$ and $S_{ij}$ are the components of the second fundamental form with respect to the basis $(\partial_1f, \dots, \partial_{n-1}f)$.
If $\tau = a^k\partial_kf + bN$ satisfies $S_{ij}\tau = \partial^2_{ij}f$, then
$$
S_{ij}a^k\partial_kf + S_{ij}bN = \Gamma^k_{ij}\partial_k f + S_{ij}N.
$$
This holds if and only if $b = 1$ and $a^kS_{ij}=\Gamma^k_{ij}$ for any $1 \le i,j,k\le n-1$. In general, there is no such relationship between the second fundamental form $S_{ij}$ and the Christoffel symbols $\Gamma^k_{ij}$.
Om the other hand, if at a point $p$, $\partial_k\phi_{ij} = 0$, then the formula for $\Gamma^k_{ij}$ in terms of $\phi^{ab}, \partial_c\phi_{ab}$ impplies that $\Gamma^k_{ij}(p) = 0$. In that case, you can just set $\tau = N$.
