Like algebraic geometry differential geometry is a notoriously bard subject to teach and to 'selfstudy.
Partly because it is very large and it is not easy to select a coherent basic chunk, partly because it is well developed and advanced.
On the other hand the subject is of vast importance in terms of applications especially to modern physics.

The authors have some 15 years experience in teaching a coherent course on the topic covering all the essentials. This volume is the distilled essence of their course.
Mathematics is a tool for thought A highly necessary tool in a world where both feedback and non-linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences.
Read more at http://ebookee.org/-quot-Basic-Elements-of-Differential-Geometry-and-Topology-quot-by-S-P-Novikov-A-T-Fomenko_1681603.html#YoZcMbMxy60SjV4R.99