Bhaskaracharya Pratishthana
is pleased to announce the third lecture
by Prof. Ravindra Kulkarni
Title: Euler-Gauss-Bonnet Theorem over K, an arbitrary subfield of the field of real numbers
Details of the Lecture:
路 Date: Monday, 14th July 2025
路 Time: 4:00 PM to 5:00 PM
路 Venue: AV Hall, Library Building, Bhaskaracharya Pratishthana, Pune
路 Speaker: Prof. Ravindra Kulkarni, Bhaskaracharya Pratishthana, Pune
路 Abstract: Let E^2_K be the Euclidean plane over K, a subfield of R, the field of real numbers. For Euclid, an "angle" was a "spatial" notion: namely, an angle is the region "between" two rays lying in E^2_K. We shall carefully define an oriented angle, an obtuse, acute, and right angles, a straight angle, a whole angle, and a sum of angles in E^2_K Later, we shall construct the space of "generalized angles", which mimics the construction of the Riemann surface of log z. Let P_n be a piecewise linear, positively oriented, polygon with n sides, n \ge 3, in E^2_K. We assume that its boundary is a simple closed curve. The "curvature" of P_n is concentrated at its vertices. This "curvature" is not a number, but it is an "angle". It is the zero angle everywhere, except at the vertices. At a vertex, the curvature of P_n is the "external angle" at the vertex. The "total curvature" of P_n is the "sum" of these angles. The basic "Euler-Gauss-Bonnet Theorem over K" is that the total curvature of P_n is W, the whole angle. What is remarkable is that it is independent of n. In this and probably the next few lectures, we shall recall the important topological notion of the Euler characteristic of a finite simplicial complex, or a finite CW-complex, and define a "polygonal surface over K with an Euclidean structure", and prove the general "Euler-Gauss-Bonnet Theorem over K" for such surfaces.
All are cordially invited to attend the lecture.