First Steps in Differential Geometry Undergraduate Texts in Mathematics

the long version

Introduces symplectic and contact geometry alongside Riemannian geometry, unlike other texts in differential geometry. Develops tools from linear algebra and advanced calculus, including differential forms and tensors, that are necessary in differential geometry. Introduces the reader to higher mathematics, including proofs of most of the main statements and results . Aimed as a text for undergraduate students who have finished two years of standard mathematics curriculum, including courses in calculus, linear algebra, and differential equations. Differential geometry arguably offers the smoothest transition from the standard university mathematics sequence of the first four semesters in calculus, linear algebra, and differential equations to the higher levels of abstraction and proof encountered at the upper division by mathematics majors. Today it is possible to describe differential geometry as "the study of structures on the tangent space," and this text develops this point of view. This book, unlike other introductory texts in differential geometry, develops the architecture necessary to introduce symplectic and contact geometry alongside its Riemannian cousin. The main goal of this book is to bring the undergraduate student who already has a solid foundation in the standard mathematics curriculum into contact with the beauty of higher mathematics. In particular, the presentation here emphasizes the consequences of a definition and the careful use of examples and constructions in order to explore those consequences -- Publisher's website.