Eschrig H. Topology and Geometry for Physics

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Outer

symmetry of space–time, 259

vector, 127

Paley–Wiener theorem, 38

Paracompact space, 34

Parallel

section, 231

tensor ﬁeld, 237

transport, 220, 231

Parametrized curve, 60

Partition of unity, 34

Pathwise connected, 42

component, 41

Period, 134

Pfafﬁan equation system, 80

Phase space, 83, 113

Physics of vibrations, 21

Poincaré invariant, 114

Poincaré’s duality, 145

Point of closure, 12

Poisson bracket, 84

Polarization, 272

Polyhedron, 141

Pontrjagin class, 274

Positive deﬁnite rank 2 tensor, 300

Principal

curvature, 323

ﬁber bundle, 206

Product

Exterior, 70, 102

Lie, 68

manifold, 60

Semi-direct, 197

Tensor, 20, 97, 99

topology, 13

Proper time, 327, 329

Pseudo-group, 56

Pseudo-Riemannian

geometry, 154, 308

Pseudo-tensor, 121

Pseudo-tensorial r-form, 222

Pull back, 72

Pure gauge potential, 262

Push forward, 71,

Quantum Hall effect, 289–292

Quantum states, 22

Quaternions, 200

Quotient

space, 16

topology, 39

Raising and lowering of tensor indices, 303

Rapidly decaying functions, 37

Reduced ﬁber bundle, 209

Reducible, 209

Reduction of the structure group, 209

Reduction theorem, 225

Reﬂexive space, 19

Regular

n-simplex, 127

domain, 127

embedding, 76

topological space, 32

F-related tangent vector ﬁelds, 78

Relative

homology, 151

topology, 12

Relatively compact, 30

Representation, 177

regular, 209

Ricci tensor, 332

Riemannian

connection, 308

geometry, 308

manifold, 300

Generalized, 300

Homogeneous, 306

metric, 301

Indeﬁnite, 301

structure, 308

surface, 337

vector bundle, 308

Rigid body, 307

Saddle point, 149

Schauder’s ﬁxed point theorem, 30

Schwarz inequality, 19

Second countable, 13

Section, 210, 227

Global, 210

Local, 210, 227

Canonical, 216

parallel, 231

Sectional curvature,

322

Semi-direct product, 197

Seminorm, 17

Separable, 12, 16

Sesquilinear function, 19

Compendium 387

S (cont.)

Set, 11

Short exact sequence, 132

r-simplex, 123

Singular, 124

Standard, 123

Simply connected, 47

Singular

r-simplex, 124

point of a tangent vector ﬁeld, 77, 255

Skeleton, 143

Skyrmion, 51

Smooth, 26

bundle, 205

manifold, 55, 58

mapping, 60

real function, 67

tangent vector ﬁeld, 68

Space

Base, 36, 206, 227

Constant curvature, 326

Homogeneous, 210

Span, 16

Special

linear group, 193

orthogonal group, 195

unitary group, 195

Standard

r-simplex, 123

horizontal vector ﬁeld, 220

Star operator, 121, 155

Stokes’ theorem, 119, 126, 129

Strict contraction, 14

Strong Morse inequality, 153

Structure

Almost complex, 341

Complex, 341

constants, 175

equations, 176, 223, 234

group, 206, 227

Hermitian, 339, 340

Subalgebra,

179

Subbundle, 209

Submanifold

Closed, 75

Embedded, 75

Open, 75

Subordinate, 34

Sum of vector bundles, 228

Support

of a distribution, 37

of a function, 33

Surjective, xi

Symmetric tensor, 101

Symplectic

K-group, 196

group, 196

structure, 113

Tangent

map, 71

space, 64

vector, 64

vector ﬁeld, 67

Complete, 81

Singular point of a, 77, 255

Smooth, 68

Taylor expansion, 24

Tempered distributions, 37

Tensor

algebra, 98

Alternating, 102

contraction, 99

ﬁeld, 107

Parallel, 237

ﬁelds,

Algebra of, 107

Fundamental, 154, 300

Levi-Civita pseudo-, 155, 331

Metric, 107, 154, 300

product, 20, 97, 99

of vector bundles, 228

Pseudo-, 121

rank, 303

Ricci, 332

space, 98

Symmetric, 101

Tensorial r

-form, 222

Theorema egregium, 324

Thermodynamic limit, 162

Thermodynamics, 94

Tietze’s extension theorem, 33

Todd class, 273

Topological

charge, 48

dual, 18

invariant, 13

space, 11

vector space, 15

Topology, 11

Coarser, 11

Discrete, 11

Finer, 11

Product, 13

Quotient, 39

Relative, 12

388 Compendium

Trivial, 11

Weak, 18

Weak*, 32

Torsion

form, 234

operation, 238

tensor ﬁeld, 238, 344

Torus group, 173

Total

curvature, 323

derivative, 23

Totally disconnected, 39

Transformation, 81, 100

Transition function, 57, 207

Translation, 174

Tychonoff’s ﬁxed point theorem, 30

Unit cell of the crystal lattice, 161

Unitary

group, 193

operator, 19

space, 19

Universal

covering group, 185, 188

covering space, 184

Urysohn’s theorem, 33

van Hove

singularities, 164

Vector

Contravariant, 64, 99

Cotangent, 66

Covariant, 64, 99

ﬁeld, 227

Fundamental, 210

Left invariant, 174

tangent, 64

Vertical

automorphism, 209

component, 213

space, 214, 231

Volume form, 116, 317

Wandering, 326

Weak

Morse inequality, 152

topology, 18

Weak* topology, 32

Weierstrass theorem, 29

Weil homomorphism, 272

Whitney sum, 228

Compendium 389