Differential Geometry of Surfaces in Higher Dimensions

Question

I am looking for a reference that discusses "surfaces in higher dimensions." Specially I need a book or a paper about analogue of fundamental forms in $\mathbb{R}^4$ and other higher dimensions.

Thank you inadvance for your help.

Answer 1

Geometry of Submanifolds by Bang-Yen Chen, (1973, M. Dekker, New York).

I know this is a late answer, but I recently have taken a look at this book. It discusses surfaces in $\mathbb{R}^4$ so it might be an interesting reference.

Answer 2

A classic and widely accepted general reference on this subject is:

Foundations of Differential Geometry, vol. II, by Kobayashi and Nomizu.

I would have a look at Chapter 7, especially section 5, p, 29.