The Lecture Notes here is a short version of my book 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
Author(s): Gabriel Lugo
Edition: Lecture Notes Ed. 2 Draft
Publisher: University of North Carolina at Wilmington
Year: 2022
Language: English
Commentary: No attempt at file size reduction. From: http://people.uncw.edu/lugo/COURSES/DiffGeom/index.htm
Tags: Differential Geometry; Mathematical Physics
Cover
Preface
1 Vectors and Curves
1.1 Tangent Vectors
1.2 Differentiable Maps
1.3 Curves in R3
1.3.1 Parametric Curves
1.3.2 Velocity
1.3.3 Frenet Frames
1.4 Fundamental Theorem of Curves
1.4.1 Isometries
1.4.2 Natural Equations
2 Differential Forms
2.1 One-Forms
2.2 Tensors
2.2.1 Tensor Products
2.2.2 Inner Product
2.2.3 Minkowski Space
2.2.4 Wedge Products and 2-Forms
2.2.5 Determinants
2.2.6 Vector Identities
2.2.7 n-Forms
2.3 Exterior Derivatives
2.3.1 Pull-back
2.3.2 Stokes' Theorem in Rn
2.4 The Hodge Operator
2.4.1 Dual Forms
2.4.2 Laplacian
2.4.3 Maxwell Equations
3 Connections
3.1 Frames
3.2 Curvilinear Coordinates
3.3 Covariant Derivative
3.4 Cartan Equations
4 Theory of Surfaces
4.1 Manifolds
4.2 The First Fundamental Form
4.3 The Second Fundamental Form
4.4 Curvature
4.4.1 Classical Formulation of Curvature
4.4.2 Covariant Derivative Formulation of Curvature
4.5 Fundamental Equations
4.5.1 Gauss-Weingarten Equations
4.5.2 Curvature Tensor, Gauss's Theorema Egregium
5 Geometry of Surfaces
5.1 Surfaces of Constant Curvature
5.1.1 Ruled and Developable Surfaces
5.1.2 Surfaces of Constant Positive Curvature
5.1.3 Surfaces of Constant Negative Curvature
5.1.4 Bäcklund Transforms
5.2 Minimal Surfaces
5.2.1 Minimal Area Property
5.2.2 Conformal Mappings
5.2.3 Isothermal Coordinates
5.2.4 Stereographic Projection
5.2.5 Minimal Surfaces by Conformal Maps
6 Riemannian Geometry
6.1 Riemannian Manifolds
6.2 Submanifolds
6.3 Sectional Curvature
6.4 Big D
6.4.1 Linear Connections
6.4.2 Affine Connections
6.4.3 Exterior Covariant Derivative
6.4.4 Parallelism
6.5 Lorentzian Manifolds
6.6 Geodesics
6.7 Geodesics in GR
6.8 Gauss-Bonnet Theorem
References
Index
