Curvature in Mathematics and Physics

Name: Curvature in Mathematics and Physics
Brand: Dover Publications
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Shlomo Sternberg(Autor*in)

Dover Publications (Verlag)

Erschienen am 17. April 2013

770 Seiten

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Inhalt

Cover Page
Title Page
Copyright Page
0.1 Introduction
Contents
1.1 Volume of a thickened hypersurface
1.2 Defining some of our terms
1.3 The Gauss map and the Weingarten map
1.4 Proof of the volume formula
1.5 Gauss's theorema egregium
1.5.1 First proof, using inertial coordinates
1.5.2 Second proof. The Brioschi formula
1.6 Back to the area formula
1.6.1 An alternative expression for the surface area
1.6.2 The mean curvature and minimal surfaces
1.6.3 Minimal hypersurfaces
1.7 Problem set - Surfaces of revolution
1. Gauss's theorema egregium.
2.1 Superalgebras
2.2 Differential forms
2.2.1 Linear differential forms
2.2.2 O(M), the algebra of exterior differential forms
2.3 The d operator
2.4 Even and odd derivations of a superalgebra
2.5 Pullback
2.6 Chain rule
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 Forms as multilinear functions on vector fields
2.14 Problems
2.14.1 Matrix valued differential forms
2.14.2 Actions of Lie groups on themselves
2.14.3 The Maurer-Cartan form
2.14.4 The Maurer-Cartan equation(s)
2.14.5 Restriction to a subgroup of Gl(n)
2.14.6 Interlude, the Haar integral
2.14.7 Frames
2.14.8 Euclidean frames
2.14.9 Frames adapted to a submanifold
2.14.10 Curves and surfaces - their structure equations
2.14.11 The sphere as an example
2.14.12 Ribbons
2.14.13 The induced metric and the Weingarten map determine a surface up to a Euclidean motion
2.14.14 Back to ribbons
2.14.15 Developing a ribbon
2.14.16 The general Maurer-Cartan equations
2.14.17 Back to curves in R3
2.14.18 Summary in more logical order, starting from Weil's formula
2. Rules of calculus.
3.1 Definition of a linear connection on the tangent bundle
3.2 Christoffel symbols
3.3 Parallel transport
3.4 Parallel vector fields along a curve
3.5 Geodesics
3.5.1 Making a linear change of parameter
3.6 Torsion
3.7 Curvature
3.7.1 R is a tensor
3.7.2 The first Bianchi identity
3.8 Tensors and tensor analysis
3.8.1 Multilinear functions and tensors of type (r, s)
3.8.2 Tensor multiplication
3.8.3 Why a tensor over V(M) is a "field" of tensors over M
3.8.4 Tensor notation
3.8.5 Contraction
3.8.6 Tensor derivations
3.8.7 A connection as a tensor derivation, covariant differential
3.8.8 Covariant differential vs. exterior derivative
3.9 Variations and the Jacobi equations
3.9.1 Two parameter maps
3.9.2 Variations of a curve
3.9.3 Geodesic variations and the Jacobi equations
3.9.4 Conjugate points
3.10 The exponential map
3.10.1 The differential of the exponential map at 0 is the identity
3.10.2 Normal neighborhoods
3.10.3 Normal coordinates
3.10.4 The exponential map and the Jacobi equation
3.10.5 Polar maps
3.11 Locally symmetric connections
3.12 Normal neighborhoods and convex open sets
3. Connections on the tangent bundle.
4.1 Isometric connections
4.2 Levi-Civita's theorem
4.3 The Christoffel symbols of the Levi-Civita connection
4.4 Geodesics in orthogonal coordinates
4.5 The hereditary character of the Levi-Civita connection
4.6 Back to the isometric condition ?g = 0
4.7 Problems: Geodesics in the Schwarzschild exterior
4.7.1 The Schwarzschild solution
4.7.2 Massive particles
4.7.3 Orbit Types
4.7.4 Perihelion advance
4.7.5 Massless particles
4.7.6 Kerr-Schild form
4.7.7 A brief biography of Schwarzschild culled from wikipedeia and St. Andrews
4.8 Curvature identities
4.9 Sectional curvature
4.9.1 Degenerate and non-degenerate planes
4.9.2 Definition of the sectional curvature
4.9.3 The sectional curvature determines the Riemann curvature tensor
4.9.4 Constant curvature spaces
4.10 Ricci curvature
4.11 Locally symmetric semi-Riemannian manifolds
4.11.1 Why the word "symmetric" in locally symmetric?
4.12 Curvature of the induced metric of a submanifold
4.12.1 The case of a hypersurface
4.13 The de Sitter universe and its relatives
4.13.1 The Einstein field equations with cosmological constant ?
4.13.2 Cosmological considerations
4. Levi-Civita's theorem.
5.1 The Lie algebra of a Lie group
5.2 The general Maurer-Cartan form
5.3 Left invariant and bi-invariant metrics
5.4 Geodesics are cosets of one parameter subgroups
5.5 The Riemann curvature of a bi-invariant metric
5.6 Sectional curvatures
5.7 The Ricci curvature and the Killing form
5.8 Bi-invariant forms from representations
5.9 The Weinberg angle
5. Bi-invariant metrics on a Lie group.
6.1 Frame fields and coframe fields
6.1.1 The tautological tensor
6.2 Connection and curvature forms in a frame field
6.2.1 Connection forms
6.2.2 Cartan's first structural equation
6.2.3 Symmetry properties of ?
6.2.4 Curvature forms in a frame field
6.2.5 Cartan's second structural equation
6.2.6 Both structural equations in compact form
6.3 Cartan's lemma and Levi-Civita's theorem
6.3.1 Cartan's lemma in exterior algebra
6.3.2 Using Cartan's lemma to prove Levi-Civita's theorem
6.4 Examples of Cartan style computations
6.4.1 Polar coordinates in two dimensions
6.4.2 Hyperbolic geometry
6.4.3 The Schwarzschild metric
6.5 The second Bianchi identity
6.6 A theorem of F. Schur
6.7 Friedmann Robertson Walker metrics
6.7.1 The expanding universe and the big bang
6.8 The rotating black hole
6.8.1 Killing fields and Noether's theorem
6.8.2 The definition of the Kerr metric and some of its elementary properties
6.8.3 Checking that we do have a semi-Riemannian metric
6.8.4 The domains and the signature
6.8.5 An orthonormal frame field and its coframe field
6.8.6 The connection forms
6.8.7 The curvature
6. Cartan calculations
7.1 Geodesics locally minimize arc length in a Riemannian manifold
7.2 Gauss's lemma
7.3 Short enough geodesics give an absolute minimum for arc length
7. Gauss's lemma.
8.1 Jacobi fields in semi-Riemannian geometry
8.1.1 Tangential Jacobi fields
8.1.2 Perpendicular Jacobi fields
8.1.3 Decomposition of a Jacobi field into its tangential and perpendicular components
8.2 Variations of arc length
8.2.1 The first variation
8.3 Geodesics are stationary for arc length
8.3.1 Piecewise smooth variations
8.4 The second variation
8.4.1 Synge's formula for the second variation
8.5 Conjugate points and the Morse index
8.5.1 Non-positive sectional curvature means no conjugate points
8.6 Synge's theorem
8.7 Cartan on the existence of closed geodesics
8. Variational formulas.
9.1 Riemannian distance
9.1.1 Some history
9.1.2 Minimizing curves
9.2 Completeness and the Hopf-Rinow theorem
9.2.1 The key proposition - de Rham's proof
9.2.2 Geodesically complete manifolds
9.2.3 The Hopf-Rinow theorem
9.3 Hadamard's theorem
9.4 Locally isometric coverings
9.5 Symmetric spaces
9. The Hopf-Rinow theorem.
10.1 Sectional curvature and distance, locally
10.2 Myer's theorem
10.2.1 Back to the Ricci tensor
10.2.2 Myer's theorem
10.3 Length variation of a Jacobi vector field
10.3.1 Riemann's formula for the metric in a normal neighborhood
10.4 The Ricci tensor and volume growth
10. Curvature, distance, and volume.
11.1 Two dimensional Lorentz transformations
11.1.1 Two dimensional Minkowski spaces
11.1.2 Addition law for velocities
11.1.3 Hyperbolic angle aka "rapidity"
11.1.4 Proper time
11.1.5 Time dilation
11.1.6 The Lorentz-Fitzgerald contraction
11.1.7 The reverse triangle inequality
11.1.8 Physical significance of the Minkowski distance
11.1.9 Energy-momentum
11.1.10 Psychological units
11.1.11 The Galilean limit
11.2 Minkowski space
11.2.1 The Compton effect
11.2.2 Natural Units
11. Review of special relativity.
12.1 Definition of the star operator
12.1.1 The induced scalar product on exterior powers
12.2 Does *: ?kV
?n-kV determine the metric?
12.3 The star operator on forms
12.3.1 Some equations of mathematical physics
12.4 Electromagnetism
12.4.1 Two non-relativistic regimes
12.4.2 Maxwell's equations
12.4.3 Natural units and Maxwell's equations
12.4.4 The Maxwell equations with a source term
12.5 The London equations
12.5.1 The London equations in relativistic form
12.5.2 Comparing Maxwell and London
12. The star operator and electromagnetism.
13.1 Preliminaries to the preliminaries
13.1.1 Densities and n-forms
13.1.2 Densities of arbitrary order
13.1.3 Pullback of a density under a diffeomorphism
13.1.4 The Lie derivative of a density
13.1.5 The divergence of a vector field relative to a density
13.2 Divergence on a semi-Riemannian manifold
13.3 The Lie derivative of a semi-Riemannian metric
13.4 The divergence of a symmetric tensor field
13.4.1 The meaning of the condition div T = 0
13.4.2 Generalizing the condition of the vanishing of the covariant divergence
13.5 Analyzing the condition l (Lvg) = 0
13.5.1 What does condition (13.16) say for a tensor field concentrated along a curve?
13.6 Three different characterizations of a geodesic
13.7 The space of connections as an affine space
13.8 The Levi-Civita map and its derivative
13.8.1 The Riemann curvature and the Ricci tensor as a maps
13.9 An important integral identity
13. Preliminaries to the Einstein equations.
14.1 The structure of physical laws
14.1.1 The Legendre transformation
14.1.2 Inverting the Legendre transformation as the "source equation" of physics
14.2 The Newtonian example
14.3 The passive equations
14.4 The Hilbert "function"
14.5 Harmonic maps as solutions to a passive equation
14.6 Schrodinger's equation as a passive equation
14. Die Grundlagen der Physik.
15.1 The Frobenius theorem
15.1.1 Differential systems
15.1.2 Foliations, submersions, and fibrations
15.1.3 The vector fields of a differential system, the Frobenius theorem
15.1.4 Connected and maximal leaves of an integrable system
15.2 Maps into a Lie group
15.2.1 Applying the above to the diagonal
15.2.2 The induced metric and the Weingarten map determine a hypersurface up to a Euclidean motion
15.3 Another application of Frobenius: to reduction
15.4 A dual formulation of Frobenius' theorem
15.5 Horizontal and basic forms of a fibration
15.6 Reduction of a closed form
15. The Frobenius theorem.
16.1 Connection and curvature forms in a frame field
16.2 Change of frame field
16.3 The bundle of frames
16.3.1 The form
16.3.2 The form in terms of a frame field
16.3.3 The definition of ?
16.4 Connection forms in a frame field as a pull-backs
16.5 Submersions, fibrations, and connections
16.5.1 Submersions
16.5.2 Fibrations
16.5.3 Projection onto the vertical
16.5.4 Frobenius, generalized curvature, and local triviality
16.6 Principal bundles and invariant connections
16.6.1 Principal bundles
16.6.2 Connections on principal bundles
16.6.3 Associated bundles
16.6.4 Sections of associated bundles
16.6.5 Associated vector bundles
16.6.6 Exterior products of vector valued forms
16.7 Covariant differentials and derivatives
16.7.1 The horizontal projection of forms
16.7.2 The covariant differential of forms on P
16.7.3 A formula for the covariant differential of basic forms
16. Connections on principal bundles.
17.1 A brief history of gauge theories
17.2 Cartan's approach to connections
17.2.1 Cartan connections
17.3 Symmetry breaking and "mass acquisition"
17.3.1 The Automorphism group and the Gauge group of a principal bundle
17.3.2 Mass acquisition - the Higgs mechanism
17.4 The Higgs mechanism in the standard model
17.4.1 Problems of translation between mathematicians and and physicists
17.4.2 The Higgs mechanism in the standard model
17. Reduction of principal bundles.
18.1 Superbundles
18.1.1 Superspaces and superalgebras
18.1.2 The tensor product of two superalgebras
18.1.3 Lie superalgebras
18.1.4 The endomorphism algebra of a superspace
18.1.5 Superbundles
18.1.6 The endomorphism bundle of a superbundle
18.1.7 The centralizer of multiplication by differential forms
18.1.8 Bundles of Lie superalgebras
18.2 Superconnections
18.2.1 Extending superconnections to the bundle of endomorphisms
18.2.2 Supercurvature
18.2.3 The tensor product of two superconnections
18.2.4 The exterior components of a superconnection
18.2.5 A local computation
18.3 Superconnections and principal bundles
18.3.1 Recalling some definitions
18.3.2 Generalizing the above to superconnections
18.4 Clifford Bundles and Clifford superconnections
18. Superconnections.
19.1 Submersions
19.2 The fundamental tensors of a submersion
19.2.1 The tensor T
19.2.2 The tensor A
19.2.3 Covariant derivatives of T and A
19.2.4 The fundamental tensors for a warped product
19.3 Curvature
19.3.1 Curvature for warped products
19.3.2 Sectional curvature
19.4 Reductive homogeneous spaces
19.4.1 Homogeneous spaces
19.4.2 Normal symmetric spaces
19.4.3 Orthogonal groups
19.4.4 Dual Grassmannians
19.5 Schwarzschild as a warped product
19. Semi-Riemannian submersions.
Bibliography
Index
Back Cover

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