National University of Sciences and Technology
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MATH-803 Geometry
Campus SNS
Programs PG
Session Fall Semester 2016
Course Title Geometry
Course Code MATH-803
Credit Hours 3-0
Course Objectives Euclidean Geometry had been the major driving force behind the development of mathematics from Greek times on. It led to the development of the Kinematics of Aristotle, Copernicus, Kepler and Galileo. It was then used by Fermat to initiate the development of Calculus and used much more extensively by Newton for that purpose. Despite the trend away from Geometry due to the formulation of Calculus by Leibnitz, it retained the centre stage of Mathematics for a long time. In the 17th century the critique of Euclid’s fifth axiom and the subsequent development of non-Euclidean geometry made people uneasy with Geometry till the development of Differential Geometry by Gauss and its subsequent formal extension by Riemann. Though it seems to have gone somewhat out of favour with the Mathematics community more recently, its heavy utilization in Physics has once again made it an indispensable tool for many purposes, including such diverse areas as Computational Mathematics and Theoretical Physics. In this course the subject of Differential Geometry as developed by Gauss and Riemann will be explained using Penrose’s abstract index notation where possible. Cartan’s differential forms and their use for integration on manifolds will also be discussed. The methods of Lie to use groups for studying Geometry will be explained at the end.
Detail Content The theory of space curves and surfaces, with special reference to Gauss’ formulation. Extension of the theory of space curves to higher dimensions. Manifolds. Vector fields on manifolds, the metric tensor and covariant derivatives. Intrinsic curvature by the curvature tensors and scalar. Curves on manifolds, intrinsic and Lie derivatives. Isometries and other symmetries of spaces. Differential forms and integration of scalars and tensors on manifolds. The Weyl tensor. Lie groups and Lie algebras.
Text/Ref Books Textbooks (AQ): Einstein’s General Theory of Relativity
Author: Asghar Qadir
Publisher: Draft

Textbook (AG): Introduction to Differential Geometry (Second Edition)
Author: Abraham Goetz
Publisher: Addison Wesley 1970
Time Schedule Fall Semester 2014
Faculty/Resource Person Dr. Tooba Siddiqui
PhD QAU Islamabad
Discipline: Genaral Relativity