The Gauss Equations are able to give you some coarse information immediately, see for instance the wikipedia article http://en.wikipedia.org/wiki/Gaussâ€“Codazzi_equations here. For instance, if $M$ is $n$ dimensional, and you have such an isometric embedding, then about some point $p \in M\cap \mathbb{R}^N$ there is a basis of $N-n$ vectorfields, say ${e_i}$ normal to $M$ in $R^N$. Then for each $e_i$ one gets an operator $X \to \nabla_X e_i$ so in this sense there are $N- n$ second fundamental forms. Now the gauss equation writes the curvature of $M$ in terms of the sum of these operators, so in particular if $M$ has bounded unbounded curvature, so must this sum. In particular if $N = n + 1$ then a necessary condition for an embedding (with bounded shape operatorsecond fundamental form) is that the curvature of $M$ is bounded.
The Gauss Equations are able to give you some coarse information immediately, see for instance the wikipedia article http://en.wikipedia.org/wiki/Gaussâ€“Codazzi_equations here. For instance, if $M$ is $n$ dimensional, and you have such an isometric embedding, then about some point $p \in M\cap \mathbb{R}^N$ there is a basis of $N-n$ vectorfields, say ${e_i}$ normal to $M$ in $R^N$. Then for each $e_i$ one gets an operator $X \to \nabla_X e_i$ so in this sense there are $N- n$ second fundamental forms. Now the gauss equation writes the curvature of $M$ in terms of the sum of these operators, so in particular if $M$ has bounded curvature, so must this sum. In particular if $N = n + 1$ then a necessary condition for an embedding (with bounded shape operator) is that the curvature of $M$ is bounded.