Introduction
0. Fundamental Not(at)ions
1. Sets
2. Functions
3. Physical Background
Ⅰ. Real Vector Spaces
1. Spaces
Subspace geometry, components
2. Maps
Linearity, singularity, matrices
3. Operators
Projections, eigenvMues, determinant, trace
Ⅱ. Affine Spaces
1. Spaces
Tangent vectors, parallelism, coordinates
2. Combinations of Points
Midpoints, convexity
3. Maps
Linear parts, translations, components
Ⅲ. Dual Spaces
1. Contours, Co- and Contravariance, Dual Basis
Ⅳ. Metric Vector Spaces
1. Metrics
Basic geometry and examples, Lorentz geometry
2. Maps
Isometries, orthogonal projections and complements, adjoints
3. Coordinates
Orthonormal bases
4. Diagonalising Symmetric Operators
Principal directions, isotropy
Ⅴ. Tensors and Multilinear Forms
1. Multilinear Forms
Tensor Products, Degree, Contraction, Raising Indices
Ⅵ. Topological Vector Spaces
1. Continuity
Metrics: topologies, homeomorphisms
2. Limits
Convergence and continuity
3. The Usual Topology
Continuity in finite dimensions
4. Compactness and Completeness
Intermediate Value Theorem, convergence, extrema
Ⅶ. Differentiation and Manifolds
1. Differentiation
Derivative as local linear approxiamation
2. Manifolds
Charts, maps, diffeomorphisms
3. Bundles and Fields
Tangent and tensor bundles, metric tensors
4. Components
Hairy Ball Theorem, transformation formulae, raising indic
5. Curves
Parametrisation, length, integration
6. Vector Fields and Flows
First order ordinary differential equations
7. Lie Brackets
Commuting vector fields and flows
Ⅷ. Connections and Covariant Differentiation
1. Curves and Tangent Vectors
Representing a vector by a curve
2. Rolling Without Turning
Differentiation along curves in embedded manifolds
3. Differentiating Sections
Connections horizontal vectors, Christoffel symbols
4. Parallel Transport
Integrating a connection
5. Torsion and Symmetry
Torsion tensor of a connection
6. Metric Tensors and Connections
Levi-Civita connection
7. Covariant Differentiation of Tensors
Parallel transport, Ricci's Lemma, components, constancy
Ⅸ. Geodesics
1. Local Characterisation
Undeviating curves
2. Geodesics from a Point
Completeness, exponential map, normal coordinates
3. Global Characterisation
Criticality of length and energy, First Variation Formula
4. Maxima, Minima, Uniqueness
Saddle points, mirages, Twins 'Paradox'
5. Geodesics in Embedded Manifolds
Characterisation, examples
6. An Example of Lie Group Geometry
2x2 matrices as a pseudo-Riemannian manifold
Ⅹ. Curvature
1. Flat Spaces
Intrinsic description of local flatness
2. The Curvature Tensor
Properties and Components
3. Curved Surfaces
Ganssian curvature, Gauss-Bonnet Theorem
4. Geodesic Deviation
Tidal effects in spacetime
5. Sectional Curvature
Schur's Theorem, constant curvature
6. Ricci and Einstein Tensors
Signs, geometry, Einstein manifolds, conservation equation
7. The Weyl Tensor
Ⅺ. Special Relativity
1. Orienting Spacetimes
Causality, particle histories
2. Motion in Flat Spacetime
Inertial frames, momentum, rest mass, mass-energy
3. Fields
Matter tensor, conservation
4. Forces
No scalar potentials
5. Gravitational Red Shift and Curvature
Measurement gives a curved metric tensor
Ⅻ. General Relativity
1. How Geometry Governs Matter
Equivalence principle, free fall
2. What Matter does to Geometry
Einstein's equation, shape of spacetime
3. The Stars in Their Courses
Geometry of the solar system, Schwarzschild solution
4. Farewell Particle
Appendix.Existence and Smoothness of Flows
1. Completeness
2. Two Fixed Point Theorems
3. Sequences of Functions
4. Integrating Vector Quantities
5. The Main Proof
6. Inverse Function Theorem
Bibliography
Index of Notations
Index