I have been reading "An Introduction to Riemannian Geometry" by Godinho and Natário and in this chapter they introduce the notion of the second fundamental form. Then they give an example where they start with the gauss map, wich to each point $p$ we assign a unit normal vector $n_p$. Then the authors say this wich i dont quite understand "Since n_p is normal to $T_pN$, we can identify the tangent spaces $T_pN$ and $T_{g(p)}S^n$ and obtain a well-defined map $(dg)_p : T_pN → T_pN.$"
Also im having some doubts with this subject, i understand the definiton and the concept of it and the importance its gonna have with relating curvatures of the manifold and the submanifold, but i cant quite see how we are gonna calculate it and work with it, so if anyone knows where to find some examples i would appreciate it, or if anyone knows a book that explains this in the fashion im looking for i would appreciate it, Thanks!