Differential Geometry for Physics Students
This Lecture series covers and expands upon the Differential Geometry topics in Chapters 11 and 12 of the textbook “Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein”.
This is a junior or senior undergraduate course for physics, astronomy and math majors.
The video lectures on Differential Geometry for Physics Students follow these lectures closely and could be studied together.
Supplemental Lecture 9
Planes, Spheres and Surfaces:
From Straight Lines, Triangles and Differentiation to Geodesics, Covariant Differentiation, Curvature and Holonomy
Abstract
We consider the geometry of planes, spheres and surfaces in the context of classical differential geometry. Geometric concepts which generalize from planes to spheres to surfaces are emphasized. This includes straight lines in planes which become geodesics on spheres and surfaces. Triangles in planes which become geodesic triangles on spheres and surfaces. The properties of such triangles are sensitive to the curvature of the sphere and general curved surfaces. Differentiation of functions and vectors on a plane generalize to covariant differentiation on spheres and curved surfaces. Covariant differentiation lead to constructions of quantities which measure curvature locally on spheres and surfaces. The spatial distribution of geodesics is controlled by differential equations which are sensitive to the curvature of the surfaces they are embedded in. The equations of Jacobi fields, the two dimensional analog of geodesic deviation, are obtained and studied.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Differential Geometry, curvature, geodesic, covariant differentiation, Gaussian curvature, holonomy, Jacobi fields.
Contents
Planar Geometry: Setting the Stage. 2
Geometry of Curved Surfaces. 27
The Gauss Map and Its Derivative, the Weingarten Map. 28
Illustrations. Parametrizing Surfaces and Computing their Properties. 33
Geometric Interpretation of the Gauss Map. 39
Gauss’ Theorem Egregium. Mainardi-Codazzi Consistency Equations. 41
Parallel Transport, Covariant Differentiation and the Gauss-Bonnet Theorem.. 44
Parallel Transport and Covariant Differentiation. Illustrations. 44
The Turning angle for curves on Curved Surfaces and the Gauss-Bonnet Theorem.. 49
Geodesic Polar Coordinates and the Geodesic Deviation. 53
Parallel Transport, Curvature, Holonomy and Jacobi Fields. 61