Are there any example of a closed Riemannian $n$-manifold $(M,g)$ and a symmetric bilinear form $A=h_{ij}dx^i\otimes dx^j\in\Gamma(T^\ast M\otimes T^\ast M)$ satisfying Gauss' equation \begin{eqnarray} R_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk} \end{eqnarray} and Codazzi's equation \begin{eqnarray} \nabla A\text{ is symmetric as a 3-multilinear form} \end{eqnarray} such that there is no pair of an isometric immersion $f:(M,g)\to\mathbb{R}^{n+1}$ and its unit normal vector field $\nu$ whose 2nd fundamental form is $A$ ,where $\nabla$ is the Levi Civita connection w.r.t. $g$ and $\mathbb{R}^{n+1}$ is equipped with the Euclidean metric ?
Moreover, is there an example of a closed $n$-manifold $M$ and a $C^\infty$ family of its Riemannian metrics $(g_t)_{0\leq t\leq 1}$ and an one of symmetric bilinear forms $(A_t)_{0\leq t\leq 1}\subset \Gamma(T^\ast M\otimes T^\ast M)$ such that (i) $g_t$ and $A_t$ satisfy Gauss' equation and Codazzi's equation for all $t\in[0,1]$ and that (ii) $A_0$ is the 2nd fundamental form of some isometric immersion $f:(M,g_0)\to\mathbb{R}^{n+1}$ while (iii) $A_1$ is NOT a one of any isometric immersion $f:(M,g_1)\to\mathbb{R}^{n+1}$ ?