Mathematics Colloquium

Date

Wednesday, 20 March, 4-5 pm

Venue

Ramanujan Hall

Host

Jugal Verma

speaker

Nitin Nitsure

Affiliation

Bhaskaracharya Pratishthana, Pune

Title

Curvature and driving a car

Abstract

The topic of this lecture is the differential geometry and topology of surfaces in the 3-dimensional Euclidean space. But my approach will be via the common experience of driving or traveling in a car or in an auto-rickshaw. Our everyday exposure to going along the curving roads and the up and down slopes in Powai gives us powerful tools to understand the concepts of curvature of curves on a surface, and also the Gaussian curvature of the surface itself, which involves looking up vertically from the sun-roof of the car, and doing a certain book-keeping! The experience of lurching sideways while driving up or down ramps gives the alternative view of Gaussian curvature as the product of principal curvatures. I will then give a pictorial proof of the Gauss-Bonnet theorem. The last topic will be my result (proved in 2019) on the determination of the curvature tensor of a higher dimensional Riemannian manifold via quadrilateral gaps, which arose from a question by Rajaram Nityananda for surfaces. In particular, this gives a very simple formula for Gaussian curvature in terms of driving a car.
         The lecture will be mostly understandable to first-year undergraduate students. No formal background in differential geometry will be assumed. The emphasis will be on pictorial understanding rather than on epsilon-delta rigor.