I think that the subject Submanifold Geometry can satisfy what you need. And you can find the generalization in the book: Geometry 1: Basic Idea and Concepts of Differential Geometry; Chapter 3, which can be found on the website http://en.bookfi.org/book/444417.
Now I want to give some my thoughts about it. The most generalization to talk about the question is to look at the question by considering the immersion $r:M^n\rightarrow N^{n+k}$. And we will have the Fundamental Theorem about: $r:M^n\rightarrow\mathbb R^{n+k}$ and you will find it on the book I provide.
However, there are two special things we should pay attention:
First is about curve: $r:I\rightarrow\mathbb R^n$. The curve's Fundemental Theorem does only rely on curvature and its uniqueness is under the natural parameter.
Second is about hypersurface $r:M^n\rightarrow\mathbb R^{n+1}$, like surface in $\mathbb R^3$. Because it is without torsion, then we can drop Ricci Equation. So the Fundemental Theorem is only depend on The Second Fundamental Form.
