Course In Differential Geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus and integral calculus, as well as linear algebra and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology, and to the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field. Modern differential geometers have learned to present much of the subject without constant direct reference to locally defined objects that depend on a choice of coordinates. This is called the "invariant" or "coordinate free" approach to differential geometry. The only way to really see exactly what this all means is by diving in and learning the subject. Knowledge of differential geometry is common among physicists thanks to the success of Einstein's highly geometric theory of gravitation and also because of the discovery of the differential geometric underpinnings of modern gauge theory and string theory. It is interesting to note that the gauge field concept was introduced into physics within just a few years of the time that the notion of a connection on a fibre bundle was making its appearance in mathematics. This textbook gives all that is likely to be required at the undergraduate level and most of the material has in fact been taught to undergraduate.