The First Adventures on Differential Geometry

Contents
Preface
1 Differentials on Surfaces
	1.1 Introduction
	1.2 Euclidean Metric in Rn
	1.3 Functions in Rn
	1.4 Surfaces
		1.4.1 Euclidean geometry in Rn is insufficient for modeling functions on surfaces
		1.4.2 Euclidean geometry is insufficient for performing differentiations of functions on surfaces
		1.4.3 Curvatures and other properties on surfaces
	1.5 How to Use This Book
		1.5.1 Further readings
	1.6 Adventures of Beng on Differential Geometry
2 Mathematical Preliminaries
	2.1 Coordinate Representation
	2.2 Coordinate Transformation in R2
		2.2.1 Einstein’s double indices summation notation
		2.2.2 Vectors and components
	2.3 We Want Objects That Transform According to Coordinate Transformation. Why?
		2.3.1 Coordinate transformation in n dimension
	2.4 Summary
3 Fields, Derivatives and Curvilinear Coordinates
	3.1 Fields and Derivatives
		3.1.1 Scalar fields in one-dimensional space
		3.1.2 Scalar fields in two-dimensional space
		3.1.3 Derivatives
		3.1.4 Vector fields
	3.2 Nonlinear Coordinate Transformation
		3.2.1 Cartesian to polar coordinates
	3.3 Christoffel Symbols in Polar Coordinates
	3.4 Calculating Path Lengths
		3.4.1 How does the position vector trace through polar coordinates?
	3.5 Summary
4 Introduction to Curved Spaces Embedded in Rn
	4.1 Understanding a Surface in Rn
		4.1.1 Scalar and vector fields in M
		4.1.2 Concept of parameterization
	4.2 Coordinate System for Points in M
		4.2.1 Scalar functions on M and their differentials
	4.3 Tangent Vectors
		4.3.1 How to build basis vectors in a tangent space?
		4.3.2 Tangent vector fields
		4.3.3 Tangent vectors and parametric path
	4.4 Induced Metric Tensor for M ∈ Rn
		4.4.1 Calculating the path length of equator in three ways
	4.5 The Christoffel Symbols in Curved Space
		4.5.1 General equation for Christoffel symbols
	4.6 Geodesics
		4.6.1 Shortest path by the “no turning” method
	4.7 Summary
		4.7.1 Christoffel symbols and second fundamental form
5 Covariant Derivatives and Parallel Transport on M ⊂ Rn
	5.1 Covariant Derivatives in Rn
	5.2 Covariant Derivative in M
	5.3 Important Properties of Covariant Derivatives
		5.3.1 Covariant derivative on scalar functions
		5.3.2 Covariant derivative as an operator
	5.4 Covariant Derivative on a Path
	5.5 Christoffel Symbols and Their Transformation
	5.6 Parallel Transport
		5.6.1 Properties of parallel transport
		5.6.2 Exponential map
		5.6.3 Scaling properties of geodesics
	5.7 Summary
6 Intrinsic Geometry
	6.1 A Summary of Extrinsic Geometry
	6.2 From Extrinsic to Intrinsic Geometry
		6.2.1 Tangent vectors for intrinsic geometry
		6.2.2 Abstract definitions of tangent vectors
	6.3 Covectors
		6.3.1 Covector basis
		6.3.2 Abstract definition of covectors
		6.3.3 Covectors in extrinsic geometry
		6.3.4 Covariant derivative of covectors
	6.4 The Metric Tensor Is What We Need
		6.4.1 Tensor product representation of the metric tensor
		6.4.2 Covariant derivative of the metric tensor
		6.4.3 Christoffel symbols in intrinsic geometry
		6.4.4 Normal coordinates
	6.5 Pullback and Pushforward
		6.5.1 Maps between manifolds
		6.5.2 Pushforward
		6.5.3 Pushforward and covectors
		6.5.4 Properties of pushforward
		6.5.5 Pushforward for the case of extrinsic geometry
		6.5.6 Pullback
		6.5.7 Pullback of a tensor
	6.6 Summary
		6.6.1 Christoffel symbols
		6.6.2 Pushforward and covectors
7 The Lie Derivative
	7.1 Flow Fields
		7.1.1 Properties of flow fields
		7.1.2 Pushforward and flows
	7.2 The Lie Derivative of a Vector
		7.2.1 Evaluation of pushforward
	7.3 Lie Derivative of Tensors
		7.3.1 Pullback of covectors
		7.3.2 Lie derivative of covectors using Leibniz rule
	7.4 Lie Derivative of the Metric Tensor and Killing Fields
		7.4.1 Metric tensor as tensor product
		7.4.2 Lie derivative of the metric tensor
		7.4.3 Lie derivative of the metric tensor using Leibniz rule
		7.4.4 Killing fields
	7.5 The Lie Bracket
		7.5.1 Properties of the Lie bracket and Lie derivative
		7.5.2 Commuting flows and the Lie bracket
	7.6 Summary
8 Riemann Curvature Tensor
	8.1 Riemann Curvature Tensor Using Parallel Transport
		8.1.1 Parallel transport equations
		8.1.2 Path definitions and area
		8.1.3 Integration of component along paths
	8.2 Riemann Curvature Tensor Using Covariant Derivative
		8.2.1 General form of the Riemann curvature tensor with covariant derivative
	8.3 Properties of Rkimj and Rkimj
	8.4 Jacobi fields
	8.5 Sectional Curvature, the Ricci Tensor, and Scalar Curvature
		8.5.1 Sectional curvature
		8.5.2 Ricci tensor
		8.5.3 Scalar curvature
		8.5.4 Schwarzschild solution for non-rotating black holes
	8.6 Summary
9 Putting It All Together
	9.1 Equations, Definitions, Lemmas and Theorems
		9.1.1 The Riemann curvature tensor
		9.1.2 Jacobi fields
		9.1.3 Sectional curvature, Ricci tensor and Ricci scalar
		9.1.4 Einstein field equations and the Schwarzschild solution
Bibliography
Index