ISL Differential Geometry (2012)

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.

Syllabus

First week

• Smooth functions in Rn, inverse function theorem and implicit function theorem; in particular, many illustrative and explicit examples in R2 and R3.
• Smooth Urysohn lemma and partition of unity, Bump functions.
• Curves in space, regular curves, arc-length, reparametrization (by arclength).
• Plane curves, winding number, Hopf’s Umlaufsatz, Four vertex theorem (or Mukhopadhyaya’s theorem), isoperimetric inequality.
• Curves in R3, curvature and torsion of a smooth curve, Frenet-Serret formulae.

Second week

• Regular surfaces in R3, coordinate neighbourhoods, tangent vectors, tangent plane, normal fields, orientability, examples of surfaces; level sets of smooth functions on R3.

• Smooth functions on manifolds, differential of a smooth function, Gauss map, Shape operator (or the Weingarten map), normal sections, principal curvatures, Gaussian and Mean curvatures.

 Third week

• Geodesics on surfaces (of revolution etc.), geodesic curvature, geodesics as shortest paths.

• Theorema Egregium, isometries of surfaces,

•  Minimal surfaces.

• Upper half-plane (or Hyperbolic plane), Poincar´e metric, isometries and geodesics of the hyperbolic plane.

• Gauss-Bonnet theorem (statement only and applications illuminating the strong interplay between the Geometry and the Topology of a closed regular surface in R3).

Speakers

1. Kaushal Verma (KV), IISc, Bangalore
2. Ravi Kulkarni (RK), IITB
3. Harish Chandra (HC), Banaras Hindu University
4. Anandteertha Mangasuli (AM), IISER Bhopal
5. Sanjay Kumar (SK), University of Delhi
6. C S Aravinda (CSA), TIFR Bangalore
7. Harish Seshadri (HS), IISc, Bangalore
8. Krishna Kaipa (KK), IITB
9. S G Dani (SGD), TIFR, Mumbai.

Unity in Mathematics Lectures:

1. S. G. Dani
2. Ravi Kulkarni

• Following applicants are shortlisted to participate in the school.
• The selected participants are requested to confirm their participation by 10th April 2012 by sending an e-mail to H S Dhami at "drhsdhami at gmail.com"or "profdhami at rediffmail.com"

      
Almora is a hill station. There are trains from Delhi up to Haldwani/Kathgodam. The names of trains are Ranikhet Express, Uttaranchal Sampark Kranti Express and AC express from Ananad Vihar in Delhi. Air service is also available for Pantnagar from Delhi but one will have to check whether the facility is available on the days of journey. Taxi shall be available from Kathgodam or Haldwani which can be arranged for the experts after knowing their itinerary.