Differential Geometry in Physics

Lectures Notes by Gabriel Lugo
University of North Carolina at Wilmington
Copyright 1995, 2004, 2020, 2021

Honor Code

I. Vectors and Curves
1.1 Tangent Vectors
1.2 Curves
1.3 Fundamental Theorem of Curves
1.4 Natural Equations
II. Differential forms
2.1 1-Forms
2.2 Tensors and Forms of Higher Rank
2.3 Exterior Derivatives
2.4 The Hodge * Operator
III. Connections
3.1 Frames
3.2 Curvilinear Coordinates
3.3 Covariant Derivatives
3.4 Cartan's Equations

Lecture Notes
(pdf 3.5 Mb)

IV. Surfaces in R
4.1 Manifolds
4.2 First Fundamental form
4.3 Second Fundamental Form
4.4 Curvature
4.5 Fundamental Equations
V. Geometry of Surfaces in R
5.1 Surfaces of constant Curvature
5.2 Minimal Surfaces
5.3 Conformal Maps
VI. Riemannian Geometry
6.1 Riemannian Manifolds
6.2 Submanifolds
6.3 Big D
6.4 Lorentzian Manifolds
6.5 Geodesics
6.6 Gauss Bonnet Theorem

Original Handwritten Notes
(pdf 3 Mb)

VII. Lie Groups
7.1 Lie Groups
7.2 Lie Algebras
VIII. Classical  Groups in Physics
8.1 Orthogonal Groups
8.2 Lorentz Group
8.3 NP-Formalism
8.4 SU(3)
IX. Fiber Bundles
9.1, Fiber Bundles
9.2 Principal Bundles
9.3 Connections on PFB
9.4 Gauge Fields


The full book is published by UNC Press and is also available in Amazon.

ISBN 978-1-4696-6924-3 (cloth: alk. paper)
ISBN 978-1-4696-6925-0 (paperback: alk. paper)
Known typos and errors in original edition


Lecture Notes
(pdf 3.5 Mb)
The Lecture Notes  here is a short version which only includes the chapters covered in our one-semester course in differential geometry. In the list above, this would be chapters 1-4 and chapter 6. Thank you all for supporting higher learning


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Gabriel G. Lugo, lugo@uncw.edu
Last updated August 05, 2022