Differential Geometry studies geometrical objects using analytical methods. Like modern analysis itself, differential geometry originates in classical mechanics. For instance, geodesics and minimal surfaces are defined via variational principles and the curvature of a curve is easily interpreted as the acceleration with respect to the path length parameter.

Modern differential geometry, in its turn, strongly contributed to modern physics when, for instance, at the beginning of the twentieth century it was discovered by Einstein that a gravitational field is just a pseudo-Riemannian metric on space-time. The basic equations of gravity theory were written in terms of the curvature of a metric, which is a geometric quantity. Most recently the modern theory of elementary particles was based on gauge fields, which mathematically are connections on fiber bundles. It is closely related to differential topology, and to the geometric aspects of the theory of differential equations. Grigori Perelman’s proof of the Poincar conjecture using the techniques of Ricci flow demonstrated the power of the differential-geometric approach to questions in topology and highlighted the important role played by the analytic methods. Differential geometry of surfaces already captures many of the key ideas and techniques characteristic of the field.

In this school we will attempt to give an introduction to the basics of differential geometry through a thorough discussion of curves and surfaces, focussing mainly on teacher training and enabling an interaction between teachers and researchers working in various aspects of Differential Geometry. Kumaun University is the oldest university of this region and is one of the few universities where differential and Riemannian geometries are well represented in undergraduate and post-graduate courses. We hope this school will give a new direction and perspective in teaching of this subject as well as motivate the teachers of remote areas even more.