illustrative book of differential geometry

Question

I look for a book of differential geometry which emphasizes the illustration of the modern basic concepts and theorems and their "geometric aspects" (such as smooth manifolds, differential forms, coordinatefree Integration, orientation, geodesics, Riemannian metrics, curvature,...) to examples rather than developing new concepts and theorems. Especially (counter)examples who may motivated the modern basics are my key interest.

If you know some algebraic geometry literature, I look for a similar book as "Geometry of Schemes" by Eisenbud and Harris but now for differential geometry.

Answer 1

try these:

Differential Geometry - Clifford Taubes

Introduction to Smooth Manifolds - John Lee

these two are my favorite.