Schaum's Outline of Differential Geometry

This book cover the basics, curvature, tensors etc., touches a little bit on manifolds. For modern differential geometry topics like fibres, bundles you have to consult other books. But great for getting the basics under your belt!

The book uses some obsolete notation but it is a good introduction and has several developed applications of differential geometry.

MML does a good job insisting on the "how" but, sometimes at the expense of the "why" : why differential geometry, why tensors etc. Some parts in his text can be unclear but are always backed by excellent figures and a load of thoroughly illustrative, solved problems. A few hints : MML's treatment of the total derivative -- which he chooses to call differential -- is obscure and the reader will find it necessary to clarify the concept by going to more fundamental expositions such as Apostol's, Kline's and Massey's... On the other hand, his treatment of tensors, although lighter than Kreyszig's, is nevertheless quite good and does the job. Finally, the Gauss-Bonnet formula and theorem, in the "intrinsic geometry" chapter, are much more comprehensive than the heavily convoluted exposition in Kreyszig's book. Also, as usual in the Schaum's outlines, too many proofs are sent to the solved-problems portion, which breaks the reading linearity...Not very efficient, in my opinion. All in all, this Schaum's outline is a good introductory complement to Kreyszig's masterpiece.