Basic Elements of Differential Geometry and Topology by S.P. Novikov, A.T. Fomenko

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.
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.
Indeed it is impossible to do or understand gauge theories for example without a solid differential geometry and topology background.