Paper recommendation for "first and second fundamental form" when the surface $S \subset \mathbb{R^{n}}$

Question

It is known that for a surface $S \subset \mathbb{R^3}$ it can be found the first and the second fundamental form.

I would like to find out if this "first and second fundamental form" can be extended for a surface $S \subset \mathbb{R^{n}}$.

If you have some papers/books I will appreciate.

Answer 1

"Elementary topics in differential geometry" by John A. Thorpe is dedicated to surfaces in $\mathbb{R}^n$. Chapter $12$ deals with curvature, and in particular with the first and second fundamental forms.

Answer 2

Chapter 6 of Do Carmo's Riemannian geometry is on Isometric Immersions, and details the construction of the fundamental forms.