Excellent book on introduction to differential manifold by Loring W. Tu.
It may be difficult for someone who is unfamiliar with any mathematical topic to begin their own self-learning on the subject. The differential geometry is no distinction. Amari’s information geometry piqued my curiosity in the topic. Amari presented the idea of information geometry, which is a geometry of parametric probability densities, in his book. It’s a fascinating book to read. However, I struggled with several of the topics addressed in the book due to my lack of experience in differential geometry. I found a set of excellent lectures by Frederic Schuller on differential geometry on Youtube. I like his presentation of the subject. But I feel, for my need, I have to look deeper than just watching his lecture. From this point, I began looking for a book that is simple enough for a novice to read, something on the level of Springer’s Undergraduate Math series. Initially, I gathered some titles from quick browsing:
The first four books are too complicated and slow for me (except for Thierry Aubin, but at the cost of less clarity), while the last is too limited (yes, the book is written for physicists), in the sense that it focuses on applications rather than addressing differential geometry’s core principles using modern differential geometry notations. Then I came upon Loring T. Wu’s ‘An introduction to manifolds’. Initially, I imagined that the style of this book would be similar to that of the first category. This couldn’t be farther from the truth. Instead, I breezed through the book (compared to those on the first category). I believe the book’s presentation is really well thought out, since, as the author said,
First, we recast calculus on $\mathbb{R}^n$ in a way suitable for generalization to manifolds. We do this by giving meaning to the symbols $dx, dy$, and $dz$, so that they assume a life of their own, as differential forms, instead of being mere notations as in undergraduate calculus. While it is not logically necessary to develop differential forms on $\mathbb{R}^n$ before the theory of manifolds—after all, the theory of differential forms on a manifold in Part V subsumes that on $\mathbb{R}^n$, from a pedagogical point of view it is advantageous to treat $\mathbb{R}^n$ separately first, since it is on $\mathbb{R}^n$ that the essential simplicity of differential forms and exterior differentiation becomes most apparent. Another reason for not delving into manifolds right away is so that in a course setting the students without the background in point-set topology can read Appendix A on their own while studying the calculus of differential forms on $\mathbb{R}^n$.