Index

C^r -Atlas, 54
p-form, 81
"Theorema Egregium", 207, 242
"generalized acceleration", 158
"invariant eigenvalue", 230
"metric equation", 130
"orthogonal vectors", 46
"orthogonal", 122
"orthonormal", 46, 122
"unit" vector, 45
"unit" vector field, 122
1-form, 63

A

Abelian gauge field, 254
active transformation, 10
addition, 1
affine connection, 93
affine geodesic, 102
affine parameter, 103
alternating, 31
alternating operation, 32
angle, 44
angular momentum operators, 108
Anti-De Sitter space, 180
antisymmetric, 20
arc length parametrization, 159
arc separation, 159

B

base space, 257
basis set, 6
Bernoulli's integral, 205
Bianchi's differential identities, 114
bilinear form, 16
bivector space, 48
boost, 125
bundle, 257

C

Cartan's approach, 113
Cartan's structural equations, 113
Cartesian charts, 123
Cartesian coordinate chart, 201
Cartesian coordinate systems, 55
Cauchy sequence, 176
chart (x, U) or a local coordinate system, 52
Christoffel symbols of the first and the second kind, 128
circular helix, 160
class C^r, 53
commutator or Lie bracket, 107
comoving, synchronous coordinate system, 194
complete, 176
completeness, 176
components, 6
components of T, 19
conformal curvature tensor, 156
conformal group, 221
conformal mapping, 216
conformal tensor, 218
confermally flat, 219
conjugate points, 176
connected, 52
connection 1-forms, 94
connection coefficients, 93
conservative, 204
continuity equation, 203
contraction, 28
contravariant (or conjugate) metric tensor, 46
contravariant order, 24
contravariant tensor, 21
coordinate or natural basis set, 108
cotangent (or covariant vector) space, 63
cotangent (or covariant) vector field, 63
covariant derivative, 93
covariant or dual vector space, 12
covariant order, 24
covariant tensor, 31
covariant vector, 11
covector, 11
critical-valued, 170
cross-section, 262
curvature 2-forms, 113
curvature invariant, 142
curvature operator, 110
curvature scalar, 142
curvature tensor, 110
curved, 112, 149

D

D'Alembertian, 156
diffeomorphism, 58
differentiable fiber bundle, 262
differentiable function, 63
dilation, 221
dimension, 6
direct sum, 30
directional covariant derivative, 96
distance function, 176
double Hodge-dual, 156
doubly-null coordinate chart, 222
down-star (_*) duality mapping, 40
dual, 11
dummy indices, 7
dyads, 14

E

Einstein space, 214
Einstein summation convention, 7
Einstein tensor, 142
elastic deformation, 202
elasticity tensor, 203
elliptic, 230
embedded, 207
embedded manifold, 233
embedding, 233
equation of state, 204
Euclidean manifold, 123
Euclidean metric, 43
Euler-Lagrange equations, 169
Euler-Poincaré characteristic, 154
exponential mapping, 182
exterior derivative, 82
exterior or Grassmann algebra, 31
exterior product, 33
extrinsic curvature, 225, 229

F

fibre, 258
fibre bundle, 258
field, 1
first contracted Bianchi's identities, 143
first fundamental form, 170, 207
first integral, 166
first normal, 161
first or principal curvature, 161
flat, 112, 149
four-potential field, 86
Frenet-Serret formulas, 161
Frenet-Serret tetrad, 164

G

gauge transformation, 86
Gauss' and Codazzi-Mainardi equations, 242
Gauss-Bonnet theorem, 154, 155
Gaussian curvature, 206
Gaussian normal, 192
generalized directional derivative, 61, 92
generalized Gauss' theorem, 152
generalized gradient, 93
generalized Hooke's law, 203
generalized Kronecker delta, 35
generalized Lorentz group, 221
generalized Lorentz transformation, 124
generalized Poincaré group, 201, 212
generalized Stoke's theorem, 88
generators of a Lie Algebra, 108
geodesic completeness, 176, 177
geodesic deviation, 171
geodesic deviation equations, 174
geodesic equations, 166, 182-184
geodesic normal, 192
geodesic surface, 207
geodesically parallel, 193
global chart, 54
grad ƒ, 61
Green's theorem, 153
group of isometry, 269

H

harmonic equation, 137
Hausdorff, 52
Hodge star operation, 49
holomorphic function, 137
holonomic basis, 108
homeomorphism, 52, 58, 258
homogeneity, 208
hybrid tensor, 235
hybrid tensors, 25
hyperbolic, 230
hyperplanes, 110
hypersurface, 188, 233
hypersurface orthogonal, 198

I

identity mixed tensors, 22
identity operator, 10
identity tensor, 100
immersion, 233
induced metric, 150
inner product, 43
integrable, 118
integral identity, 157
inversion, 221
irrotational motion, 204
isomorphic, 10
isotropic, 208

J

Jacobi field, 176
Jacobian, 54
Jacobian mapping (or derivative mapping), 73
Jacobian matrix, 74
Jordan curve, 117

K

Killing equation, 268
Killing vectors, 268
Kretschmann invariant, 213
Kretschmann scalar, 145
Kronecker delta, 7

L

Lagrangian, 169
Lame's constant, 203
Laplacian, 137
Lie bracket, 107
Lie derivative, 90, 264
Lie group, 262
line element, 170
linear combination, 5
linear form, 11
linear functional, 11
linear mapping (or transformation), 9
linear operator, 10
linear Riemannian connection, 126
linearly dependent, 5
linearly independent, 5
Lipschitz condition, 182, 263
local pseudo-group, 264
Lorentz metric, 43, 149
Lorentz transformation, 125

M

maximum, 170
mean curvature, 206
metric form, 170
metric tensor, 43
minimal submanifold, 256
minimum, 170
Minkowskian chart, 123
Minkowskian or pseudo-Cartesian coordinate system, 164
mixed second order tensor, 22
Moebius strip, 261
momentum operators, 108
multi-linearity, 24
multiplication, 1

N

negatively oriented, 41
non-degeneracy (or regularity), 64
non-degeneracy axiom, 43
non-flat or curved, 149
non-holonomic basis, 99, 108
non-Lorentz metrics, 149
non-null submanifold, 246
norm or length, 42, 122
normal or hypersurface orthogonal coordinate, 189
null hypersurface, 234
null vector, 45
numerical tensors, 30

O

orientable, 54
oriented (or pseudo) relative (r+s)-th order tensor field, 73
oriented (or pseudo) relative tensor, 41
oriented volume, 49
orthogonal, coordinate chart, 194
outer product, 17, 26

P

parabolic, 230
paracompact, 52
parallelly propagated, 101
parametrized curve, 64
parametrized submanifold, 245
passive transformation, 8

permutation (or numerical) symbol of Levi-Civita, 37
planar, 230
Poincaré's lemma, 85
positive-definite, 122
positive-definite inner product, 44
positively oriented, 41
potential equation, 137
principal roots, 230
principal, normal curvatures, 241
product bundle, 257
profile curve, 231
projection, 257
projection tensor, 255
projective curvature tensor, 120
projective transformation, 120
pseudo-Cartesian, 201
Pythagoras theorem, 171

R

raising and lowering indices, 46
raising and lowering tensor component indices, 126
real-analytic, 53
relative tensor, of weight, 39
relative tensor field, 72
reparametrization, 67, 234
reparametrized curve, 67
representation matrix, 10
Ricci identities, 120, 140
Ricci rotation coefficients, 129
Ricci tensor, 141
Ricci-flat spaces, 215
Riemann normal coordinate chart, 185
Riemann-Christoffel curvature tensor, 138, 186
Riemannian sectional curvature, 207
rigidity, 203
rotation group, 108
ruled surface, 174, 236

S

scalar density, 40
scalar product, 43
Schouten-Cotton tensor, 218
Schroedinger equation, 138
second contracted Bianchi's identities, 143
second curvature (or torsion), 161
second fundamental form, 229
second normal (or binormal), 161
second order covariant tensor, 16
separation, 44, 122, 169
signature of a metric, 44
singularity theorems, 177
skew, 31
space of constant curvature, 209
spacelike, 166
spacelike hypersurface, 234
spacelike vector, 44
special conformal transformation, 221
spherical polar coordinate system, 55
spinor, 14
spinor dyads, 9

spinor space in an arbitrary field, 38
standard basis, 6

stationary, 170
stationary space-time, 245
straight lines, 171
strain-tensor, 203
stress-tensor, 203
strict components, 37
structure coefficients, 108
structure group, 259
submanifold, 245
subtensor, 191
symmetric, 20
Symmetric group, 32
symmetric space, 213
symmetrization operation, 32
symmetrized Riemann tensor, 156

T

tangent (or contravariant) vector field, 262
tangent bundle, 262
tangent vector, 60
tangent vector space, 61
tensor algebra, 30
tensor bundle, 69, 262
tensor density, 39
tensor density field, 72
tensor field, 69
tensor product, 17, 26
tetrad, 15, 125
tetrads, 9

timelike, 166
timelike hypersurface, 234
timelike vector, 45
topological invariants, 154
topological manifold, 54
topology, 52
torsion 2-forms, 113
total partial derivative, 237
totally antisymmetric, 31
totally differentiable, 63
totally geodesic hypersurface, 244
trace operation, 28
transposition, 20, 31
triads, 15
trivial bundle, 257
typical fibre, 258

U

umbilic, 230
unit matrix, 7
up-star (*) duality mapping, 40

V

variational method, 169
vector subspace, 5

velocity potential, 204
vorticity tensor, 204

W

wave operator, 156
wave-function, 138
wave-mechanics, 138
wedge product, 33
winding number, 68
world-function, 177

Y

Yano's integral formula, 157

Z

zero mixed tensors, 22
zero or null mapping, 10
zero tensor, 16