Another free online geometry textbook written by Shlomo Sternberg from Harvard University. This text, written in 2003, stems from a one semester course, taught by the author, covering an introduction to Riemannian geometry and its principle physical application – Einstein’s theory of general relativity.

Students should have a background in linear algebra and advanced calculus with a concentration on differential forms.

Sternberg has also written the advanced mathematics textbooks – Theory of Functions of a Real Variable and Lie Algebras.

## Table of Contents

- 0.1 Introduction
- 1 The principal curvatures
- 1.1 Volume of a thickened hypersurface
- 1.2 The Gauss map and the Weingarten map
- 1.3 Proof of the volume formula
- 1.4 Gauss’s theorema egregium
- 1.4.1 First proof, using inertial coordinates
- 1.4.2 Second proof. The Brioschi formula
- 1.5 Problem set – Surfaces of revolution
- 2 Rules of calculus
- 2.1 Superalgebras
- 2.2 Differential forms
- 2.3 The d operator
- 2.4 Derivations
- 2.5 Pullback
- 2.6 Chainrule
- 2.7 Lie derivative
- 2.8 Weil’s formula
- 2.9 Integration
- 2.10 Stokes theorem
- 2.11 Lie derivatives of vector fields
- 2.12 Jacobi’s identity
- 2.13 Left invariant forms
- 2.14 The Maurer Cartan equations
- 2.15 Restriction to a subgroup
- 2.16 Frames
- 2.17 Euclidean frames
- 2.18 Frames adapted to a submanifold
- 2.19 Curves and surfaces – their structure equations
- 2.20 The sphere as an example
- 2.21 Ribbons
- 2.22 Developing a ribbon
- 2.23 Parallel transport along a ribbon
- 2.24 Surfaces in R3
- 3. Levi-Civita Connections
- 3.1 Definition of a linear connection on the tangent bundle
- 3.2 Christoffel symbols
- 3.3 Parallel transport
- 3.4 Geodesics
- 3.5 Covariant differential
- 3.6 Torsion
- 3.7 Curvature
- 3.8 Isometric connections
- 3.9 Levi-Civita’s theorem
- 3.10 Geodesics in orthogonal coordinates
- 3.11 Curvature identities
- 3.12 Sectional curvature
- 3.13 Ricci curvature
- 3.14 Bi-invariant metrics on a Lie group
- 3.14.1 The Lie algebra of a Lie group
- 3.14.2 The general Maurer-Cartan form
- 3.14.3 Left invariant and bi-invariant metrics
- 3.14.4 Geodesics are cosets of one parameter subgroups
- 3.14.5 The Riemann curvature of a bi-invariant metric
- 3.14.6 Sectional curvatures
- 3.14.7 The Ricci curvature and the Killing form
- 3.14.8 Bi-invariant forms from representations
- 3.14.9 The Weinberg angle
- 3.15 Frame fields
- 3.16 Curvature tensors in a frame field
- 3.17 Frame fields and curvature forms
- 3.18 Cartan’s lemma
- 3.19 Orthogonal coordinates on a surface
- 3.20 The curvature of the Schwartzschild metric
- 3.21 Geodesics of the Schwartzschild metric
- 3.21.1 Massive particles
- 3.21.2 Massless particles
- 4. The bundle of frames
- 4.1 Connection and curvature forms in a frame field
- 4.2 Change of frame field
- 4.3 The bundle of frames
- 4.3.1 The form θ
- 4.3.2 The form θ in terms of a frame field
- 4.3.3 The definition of ω
- 4.4 The connection form in a frame field as a pull-back
- 4.5 Gauss’ theorems
- 4.5.1 Equations of structure of Euclidean space
- 4.5.2 Equations of structure of a surface in R3
- 4.5.3 Theorema egregium
- 4.5.4 Holonomy
- 4.5.5 Gauss-Bonnet
- 5 Connections on principal bundles
- 5.1 Submersions, fibrations, and connections
- 5.2 Principal bundles and invariant connections
- 5.2.1 Principal bundles
- 5.2.2 Connections on principal bundles
- 5.2.3 Associated bundles
- 5.2.4 Sections of associated bundles
- 5.2.5 Associated vector bundles
- 5.2.6 Exterior products of vector valued forms
- 5.3 Covariant differentials and covariant derivatives
- 5.3.1 The horizontal projection of forms
- 5.3.2 The covariant differential of forms on P
- 5.3.3 A formula for the covariant differential of basic forms
- 5.3.4 The curvature is dω
- 5.3.5 Bianchi’s identity
- 5.3.6 The curvature and d2
- 6 Gauss’s lemma.
- 6.1 The exponential map
- 6.2 Normal coordinates
- 6.3 The Euler field E and its image P
- 6.4 The normal frame field
- 6.5 Gauss’ lemma
- 6.6 Minimization of arc length
- 7 Special relativity
- ￼7.1 Two dimensional Lorentz transformations
- 7.1.1 Addition law for velocities
- 7.1.2 Hyperbolic angle
- 7.1.3 Proper time
- 7.1.4 Time dilatation
- 7.1.5 Lorentz-Fitzgerald contraction
- 7.1.6 The reverse triangle inequality
- 7.1.7 Physical significance of the Minkowski distance
- 7.1.8 Energy-momentum
- 7.1.9 Psychological units
- 7.1.10 The Galilean limit
- 7.2 Minkowski space
- 7.2.1 The Compton effect
- 7.2.2 Natural Units
- 7.2.3 Two-particle invariants
- 7.2.4 Mandlestam variables
- 7.3 Scattering cross-section and mutual flux
- 8. Die Grundlagen der Physik
- 8.1 Preliminaries
- 8.1.1 Densities and divergences
- 8.1.2 Divergence of a vector field on a semi-Riemannian manifold
- 8.1.3 The Lie derivative of of a semi-Riemann metric
- 8.1.4 The covariant divergence of a symmetric tensor field
- 8.2 Varying the metric and the connection
- 8.3 The structure of physical laws
- 8.3.1 The Legendre transformation
- 8.3.2 The passive equations
- 8.4 The Hilbert “function”
- 8.5 Schrodinger’s equation as a passive equation
- 8.6 Harmonic maps
- 9 Submersions
- 9.1 Submersions
- 9.2 The fundamental tensors of a submersion
- 9.2.1 The tensor T
- 9.2.2 The tensor A
- 9.2.3 Covariant derivatives of T and A
- 9.2.4 The fundamental tensors for a warped product
- 9.3 Curvature
- 9.3.1 Curvature for warped products
- 9.3.2 Sectional curvature
- 9.4 Reductive homogeneous spaces
- 9.4.1 Bi-invariant metrics on a Lie group
- 9.4.2 Homogeneous spaces
- 9.4.3 Normal symmetric spaces
- 9.4.4 Orthogonal groups
- 9.4.5 Dual Grassmannians
- 9.5 Schwarzschild as a warped product
- 9.5.1 Surfaces with orthogonal coordinates
- 9.5.2 The Schwarzschild plane
- 9.5.3 Covariant derivatives
- 9.5.4 Schwarzschild curvature
- 9.5.5 Cartan computation
- 9.5.6 Petrov type
- 9.5.7 Kerr-Schild form
- 9.5.8 Isometries
- 9.6 Robertson Walker metrics
- 9.6.1 Cosmogeny and eschatology
- 10 Petrov types
- 10.1 Algebraic properties of the curvature tensor
- 10.2 Linear and antilinear maps
- 10.3 Complex conjugation and real forms
- 10.4 Structures on tensor products
- 10.5 Spinors and Minkowski space
- 10.6 Traceless curvatures
- 10.7 The polynomial algebra
- 10.8 Petrov types
- 10.9 Principal null directions
- 10.10 Kerr-Schild metrics
- 11 Star
- 11.1 Definition of the star operator
- 11.2 Does⋆:∧kV →∧n−kV determine the metric?
- 11.3 The star operator on forms
- 11.3.1 For R2
- 11.3.2 For R3
- 11.3.3 ForR1,3
- 11.4 Electromagnetism
- 11.4.1 Electrostatics
- 11.4.2 Magnetoquasistatics
- 11.4.3 The London equations
- 11.4.4 The London equations in relativistic form
- 11.4.5 Maxwell’s equations
- 11.4.6 Comparing Maxwell and London

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Semi-Riemann Geometry and General Relativity