Curvature in mathematics and physics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Mineola, NY
Dover Publ.
2012
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis Klappentext |
| Item Description: | Literaturverz. S. 401 und Index |
| Physical Description: | 405 S. Ill., graph. Darst. 24 cm |
| ISBN: | 0486478556 9780486478555 |
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MARC
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| 084 | |a 31.52 |2 bkl | ||
| 084 | |a MAT 530f |2 stub | ||
| 100 | 1 | |a Sternberg, Shlomo |d 1936- |e Verfasser |0 (DE-588)121101762 |4 aut | |
| 245 | 1 | 0 | |a Curvature in mathematics and physics |c Shlomo Sternberg |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Mineola, NY |b Dover Publ. |c 2012 | |
| 300 | |a 405 S. |b Ill., graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 401 und Index | ||
| 650 | 4 | |a Curvature | |
| 650 | 4 | |a Semi-Riemannian geometry | |
| 650 | 4 | |a Geometry, Differential | |
| 650 | 4 | |a Differential calculus | |
| 650 | 4 | |a Algebras, Linear | |
| 650 | 4 | |a Relativity (Physics) | |
| 650 | 0 | 7 | |a Mathematische Physik |0 (DE-588)4037952-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Differentialgeometrie |0 (DE-588)4012248-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Krümmung |0 (DE-588)4128765-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differentialgeometrie |0 (DE-588)4012248-7 |D s |
| 689 | 0 | 1 | |a Krümmung |0 (DE-588)4128765-4 |D s |
| 689 | 0 | 2 | |a Mathematische Physik |0 (DE-588)4037952-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | |m DE-601 |q pdf/application |u http://zbmath.org/?q=an:1257.53001 |y Zentralblatt MATH |3 Inhaltstext | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026819508&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026819508&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-026819508 | ||
Record in the Search Index
| _version_ | 1804151461926404096 |
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| adam_text | Contents
0.1
Introduction
.............................. 2
Gauss s
theorema
egregium.
19
1.1
Volume of a thickened hypersurface
................. 20
1.2
Defining some of our terms
...................... 21
1.3
The Gauss map and the
Weingarten map.............. 26
1.4
Proof of the volume formula
..................... 30
1.5
Gauss s
theorema
egregium
...................... 33
1.5.1
First proof, using
inerţial
coordinates
............ 36
1.5.2
Second proof. The Brioschi formula
............. 39
1.6
Back to the area formula
....................... 41
1.6.1
An alternative expression for the surface area
...... 41
1.6.2
The mean curvature and minimal surfaces
......... 42
1.6.3
Minimal hypersurfaces
.................... 43
1.7
Problem set
-
Surfaces of revolution
................. 46
Rules of calculus.
51
2.1
Superalgebras
............................. 51
2.2
Differential forms
........................... 52
2.2.1
Linear differential forms
.................... 52
2.2.2
Ω(Μ),
the algebra of exterior differential forms
....... 52
2.3
The
d
operator
............................. 53
2.4
Even and odd derivations of a superalgebra
............. 53
2.5
Pullback
................................ 55
2.6
Chain rule
............................... 55
2.7
Lie derivative
.............................. 55
2.8
Weil s formula
............................. 56
2.9
Integration
............................... 58
2.10
Stokes theorem
............................. 58
2.11
Lie derivatives of vector fields
..................... 59
2.12
Jacobi s identity
............................ 60
2.13
Forms as multilinear functions on vector fields
........... 61
2.14
Problems
................................ 62
2.14.1
Matrix valued differential forms
............... 62
2.14.2
Actions of Lie groups on themselves
............. 63
2.14.3
The Maurer-Cartan form
................... 64
2.14.4
The Maurer-Cartan equation(s)
............... 64
2.14.5
Restriction to a subgroup of Gl(n)
.............. 65
2.14.6
Interlude, the
Haar
integral
.................. 65
2.14.7
Frames
............................. 68
2.14.8
Euclidean frames
........................ 69
2.14.9
Frames adapted to a submanifold
.............. 70
2.14.10
Curves and surfaces
-
their structure equations
....... 71
2.14.11
The sphere as an example
................... 71
2.14.12Ribbons
............................ 73
2.14.13
The induced metric and the
Weingarten
map determine a
surface up to a Euclidean motion
.............. 74
2.14.14
Back to ribbons
........................ 75
2.14.15
Developing a ribbon
...................... 76
2.14.16
The general Maurer-Cartan equations
............ 77
2.14.17Back to curves in M3
..................... 78
2.14.18
Summary in more logical order, starting from Weil s for¬
mula
.............................. 81
Connections on the tangent bundle.
85
3.1
Definition of a linear connection on the tangent bundle
...... 85
3.2 Christoffel
symbols
........................... 86
3.3
Parallel transport
........................... 86
3.4
Parallel vector fields along a curve
.................. 88
3.5
Geodesies
................................ 89
3.5.1
Making a linear change of parameter
............ 90
3.6
Torsion
................................. 90
3.7
Curvature
................................ 91
3.7.1
R
is a tensor
.......................... 91
3.7.2
The first
Bianchi
identity
................... 92
3.8
Tensors and tensor analysis
...................... 93
3.8.1
Multilinear functions and tensors of type (r,s)
....... 93
3.8.2
Tensor multiplication
..................... 95
3.8.3
Why a tensor over V(M) is a field of tensors over M.
. 95
3.8.4
Tensor notation
........................ 97
3.8.5
Contraction
.......................... 98
3.8.6
Tensor derivations
....................... 99
3.8.7
A connection as a tensor derivation, covariant differential.
101
3.8.8
Covariant differential vs. exterior derivative
........ 102
3.9
Variations and the Jacobi equations
................. 104
3.9.1
Two parameter maps
.....................
ţO4
3.9.2
Variations of a curve
..................... 105
3.9.3
Geodesic variations and the Jacobi equations
........ 106
3.9.4
Conjugate points
........................ 107
3.10
The exponential map
......................... 107
3.10.1
The differential of the exponential map at
0
is the identity.
108
3.10.2 Normal
neighborhoods
.................... 108
3.10.3 Normal
coordinates
...................... 108
3.10.4
The exponential map and the Jacobi equation
....... 109
3.10.5
Polar maps
...........................
Ill
3.11
Locally symmetric connections
.................... 112
3.12
Normal neighborhoods and convex open sets
............ 112
Levi-Civita s theorem.
117
4.1
Isometric connections
......................... 117
4.2
Levi-Civita s theorem
......................... 118
4.3
The Christoffel symbols of the Levi-Civita connection
....... 120
4.4
Geodesies in orthogonal coordinates
................. 121
4.5
The hereditary character of the Levi-Civita connection
...... 122
4.6
Back to the isometric condition Vg
= 0............... 123
4.7
Problems: Geodesies in the
Schwarzschild
exterior
......... 123
4.7.1
The
Schwarzschild
solution
.................. 124
4.7.2
Massive particles
........................ 125
4.7.3
Orbit Types
.......................... 126
4.7.4
Perihelion advance
....................... 128
4.7.5
Massless particles
....................... 131
4.7.6
Kerr-Schild form
........................ 132
4.7.7
A brief biography of
Schwarzschild
culled from wikipedeia
and St. Andrews
........................ 133
4.8
Curvature identities
.......................... 135
4.9
Sectional curvature
.......................... 136
4.9.1
Degenerate and non-degenerate planes
............ 136
4.9.2
Definition of the sectional curvature
............. 137
4.9.3
The sectional curvature determines the Riemann curva¬
ture tensor
........................... 137
4.9.4
Constant curvature spaces
.................. 138
4.10
Ricci
curvature
............................. 139
4.11
Locally symmetric semi-Riemannian manifolds
........... 140
4.11.1
Why the word symmetric in locally symmetric?
.... 143
4.12
Curvature of the induced metric of a submanifold
......... 143
4.12.1
The case of a hypersurface
.................. 145
4.13
The
de
Sitter universe and its relatives
............... 145
4.13.1
The Einstein field equations with cosmological constant
Л.
148
4.13.2
Cosmological considerations
................. 148
В і-
invariant metrics on a Lie group.
151
5.1
The Lie algebra of a Lie group
.................... 151
5.2
The general Maurer-Cartan form
................... 153
5.3
Left invariant and bi-invariant metrics
................ 155
5.4
Geodesies are cosets of one parameter subgroups
.......... 156
5.5
The Riemann curvature of a bi-invariant metric
.......... 157
5.6
Sectional curvatures
.......................... 157
5.7
The Ricci
curvature and the Killing form
.............. 157
5.8
Bi-invariant forms from representations
............... 158
5.9
The
Weinberg
angle
.......................... 160
6
Cart an calculations
161
6.1
Frame fields and
córrame
fields
.................... 161
6.1,1
The tautological tensor
.................... 162
6.2
Connection and curvature forms in a frame field
.......... 162
6.2.1
Connection forms
....................... 162
6.2.2
Cartan s first structural equation
.............. 163
6.2.3
Symmetry properties of
ω
................... 163
6.2.4
Curvature forms in a frame field
............... 164
6.2.5
Cartan s second structural equation
............. 165
6.2.6
Both structural equations in compact form
......... 165
6.3
Cartan s lemma and Levi-Civita s theorem
............. 165
6.3.1
Cartan s lemma in exterior algebra
............. 166
6.3.2
Using Cartan s lemma to prove Levi-Civita s theorem.
. . 166
6.4
Examples of Cartan style computations
............... 167
6.4.1
Polar coordinates in two dimensions
............. 167
6.4.2
Hyperbolic geometry
..................... 168
6.4.3
The
Schwarzschild
metric
................... 170
6.5
The second
Bianchi
identity
...................... 172
6.6
A theorem of
F. Schur......................... 173
6.7 Friedmann
Robertson Walker metrics
................ 173
6.7.1
The expanding universe and the big bang
.......... 176
6.8
The rotating black hole
........................ 178
6.8.1
Killing fields and Noether s theorem
............. 178
6.8.2
The definition of the Kerr metric and some of its elemen¬
tary properties
......................... 179
6.8.3
Checking that we do have a semi-Riemannian metric.
. . 180
6.8.4
The domains and the signature
................ 182
6.8.5
An
orthonormal
frame field and its
córrame
field
...... 184
6.8.6
The connection forms
..................... 186
6.8.7
The curvature
......................... 187
7
Gauss s lemma.
189
7.1
Geodesies locally minimize arc length in a Riemannian manifold.
189
7.2
Gauss s lemma
............................. 190
7.3
Short enough geodesies give an absolute minimum for arc length.
192
8
Variational formulas.
193
8.1
Jacobi fields in semi-Riemannian geometry
.............
f.93
8.1.1
Tangential Jacobi fields
.................... 193
8.1.2
Perpendicular Jacobi fields
.................. 193
8.1.3
Decomposition of a Jacobi field into its tangential and
perpendicular components
.................. 194
8.2
Variations
of arc length
........................ 195
8.2.1
The first variation
....................... 195
8.3
Geodesies are stationary for arc length
............... 196
8.3.1
Piecewise smooth variations
................. 196
8.4
The second variation
......................... 197
8.4.1
Synge s formula for the second variation
........... 197
8.5
Conjugate points and the Morse index
................ 199
8.5.1
Non-positive sectional curvature means no conjugate points.
200
8.6
Synge s theorem
............................ 201
8.7
Cartan on the existence of closed geodesies
............. 202
9
The Hopf-Rinow theorem.
205
9.1
Riemannian distance
.......................... 205
9.1.1
Some history
.......................... 207
9.1.2
Minimizing curves
....................... 207
9.2
Completeness and the Hopf-Rinow theorem
............. 207
9.2.1
The key proposition
-
de Rham s
proof
........... 207
9.2.2
Geodesically complete manifolds
............... 209
9.2.3
The Hopf-Rinow theorem
................... 210
9.3
Hadamard s theorem
......................... 212
9.4
Locally isometric coverings
...................... 212
9.5
Symmetric spaces
........................... 214
10
Curvature, distance, and volume.
217
10.1
Sectional curvature and distance, locally.
.............217
10.2
Myer s theorem
............................222
10.2.1
Back to the
Ricci
tensor
...................223
10.2.2
Myer s theorem
........................225
10.3
Length variation of a Jacobi vector field
...............225
10.3.1
Riemann s formula for the metric in a normal neighborhood.
226
10.4
The
Rìcci
tensor and volume growth
.................228
11
Review of special relativity.
231
11.1
Two dimensional
Lorentz
transformations
.............. 231
11.1.1
Two dimensional Minkowski spaces
............. 231
11.1.2
Addition law for velocities
.................. 233
11.1.3
Hyperbolic angle aka rapidity
............... 234
11.1.4
Proper time
.......................... 235
11.1.5
Time dilation
......................... 235
11.1.6
The Lorentz-Fitzgerald contraction
............. 236
11.1.7
The reverse triangle inequality
................ 236
11.1.8
Physical significance of the Minkowski distance
....... 237
11.1.9
Energy-momentum
...................... 238
11.1.10
Psychological units
...................... 239
11.1.11
The Galilean limit
....................... 241
11.2
Minkowski space
............................ 241
11.2.1
The Compton effect
......................242
11.2.2
Natural Units
.........................247
12
The star operator and electromagnetism.
249
12.1
Definition of the star operator
.................... 249
12.1.1
The induced scalar product on exterior powers
....... 249
12.2
Does
* :
AfcV
->-
f n~kV determine the metric?
.......... 252
12.3
The star operator on forms
...................... 254
12.3.1
Some equations of mathematical physics
.......... 254
12.4
Electromagnetism
........................... 258
12.4.1
Two non-relativistic regimes
................. 258
12.4.2
Maxwell s equations
...................... 262
12.4.3
Natural units and Maxwell s equations
........... 263
12.4.4
The Maxwell equations with a source term
......... 264
12.5
The London equations
......................... 265
12.5.1
The London equations in relativistic form
.......... 267
12.5.2
Comparing Maxwell and London
............... 269
13
Preliminaries to the Einstein equations.
271
13.1
Preliminaries to the preliminaries
.................. 271
13.1.1
Densities and n-forms
..................... 271
13.1.2
Densities of arbitrary order
.................. 273
13.1.3
Pullback of a density under a diffeomorphism
........ 273
13.1.4
The Lie derivative of a density
................ 273
13.1.5
The divergence of a vector field relative to a density.
. . . 274
13.2
Divergence on a semi-Riemannian manifold
............. 275
13.3
The Lie derivative of a semi-Riemannian metric
.......... 278
13.4
The divergence of a symmetric tensor field
............. 278
13.4.1
The meaning of the condition
div T
= 0.......... 279
13.4.2
Generalizing the condition of the vanishing of the covari-
ant divergence
......................... 281
13.5
Analyzing the condition i(Lvs)
= 0................. 281
13.5.1
What does condition
(13.16)
say for a tensor field concen¬
trated along a curve?
..................... 283
13.6
Three
different
characterizations of a geodesic
........... 285
13.7
The space of connections as an
affine
space
............. 286
13.8
The Levi-Civita map and its derivative
............... 287
13.8.1
The Riemann curvature and the
Ricci
tensor as a maps.
. 287
13.9
An important integral identity
.................... 288
14
Die
Grundlagen der Physik. ¿91
14.1
The structure of physical laws
....................291
14.1.1
The Legendre transformation
.................291
14.1.2
Inverting the Legendre transformation as the source equa¬
tion of physics
........................292
14.2
The Newtonian example
........................292
14.3
The passive equations
......................... 294
14.4
The Hubert function
........................ 295
14.5
Harmonic maps as solutions to a passive equation
......... 299
14.6
Schrodinger s equation as a passive equation
............ 302
15
The Frobenius theorem.
303
15.1
The Frobenius theorem
........................ 304
15.1.1
Differential systems
...................... 304
15.1.2
Foliations, submersions, and fibrations
............ 305
15.1.3
The vector fields of a differential system, the Frobenius
theorem
............................. 306
15.1.4
Connected and maximal leaves of an
integrable
system.
. 308
15.2
Maps into a Lie group
......................... 309
15.2.1
Applying the above to the diagonal
............. 310
15.2.2
The induced metric and the
Weingarten
map determine a
hypersurface up to a Euclidean motion
........... 310
15.3
Another application of Frobenius: to reduction
........... 311
15.4
A dual formulation of Frobenius theorem
.............. 312
15.5
Horizontal and basic forms of a fibration
.............. 313
15.6
Reduction of a closed form
...................... 314
16
Connections on principal bundles.
315
16.1
Connection and curvature forms in a frame field
.......... 315
16.2
Change of frame field
......................... 316
16.3
The bundle of frames
......................... 318
16.3.1
The form
ů
........................... 320
16.3.2
The form
ϋ
in terms of a frame field
............. 320
16.3.3
The definition of a;
...................... 321
16.4
Connection forms in
а
frame field as a pull-backs
.......... 321
16.5
Submersions, fibrations, and connections
.............. 324
16.5.1
Submersions
.......................... 324
16.5.2
Fibrations
........................... 327
16.5.3
Projection onto the vertical
.................. 329
16.5.4
Frobenius, generalized curvature, and local triviality.
. . . 330
16.6
Principal bundles and invariant connections
............. 331
16.6.1
Principal bundles
....................... 331
16.6.2
Connections on principal bundles
.............. 334
16.6.3
Associated bundles
...................... 335
16.6.4
Sections of associated bundles
................ 336
16.6.5
Associated vector bundles
................... 337
16.6.6
Exterior products of vector valued forms
.......... 340
16.7
Covariant differentials and derivatives
................ 341
16.7.1
The horizontal projection of forms
.............. 341
16.7.2
The covariant differential of forms on
Ρ
........... 342
16.7.3
A formula for the covariant differential of basic forms.
. . 343
17
Reduction of principal bundles.
347
17.1
A brief history of gauge theories
................... 347
17.2
Cartan s approach to connections
.................. 348
17.2.1
Cartan connections
...................... 351
17.3
Symmetry breaking and mass acquisition
............. 352
17.3.1
The Automorphism group and the Gauge group of a prin¬
cipal bundle
.......................... 353
17.3.2
Mass acquisition
-
the Higgs mechanism
........... 356
17.4
The Higgs mechanism in the standard model
............ 357
17.4.1
Problems of translation between mathematicians and and
physicists
............................ 357
17.4.2
The Higgs mechanism in the standard model
........ 360
18
Superconnections.
365
18.1
Superbundles
.............................. 366
18.1.1
Superspaces and superalgebras
................ 366
18.1.2
The tensor product of two superalgebras
.......... 366
18.1.3
Lie superalgebras
....................... 367
18.1.4
The endomorphism algebra of a superspace
......... 367
18.1.5
Superbundles
.......................... 368
18.1.6
The endomorphism bundle of a superbundle
........ 368
18.1.7
The centralizer of multiplication by differential forms.
. . 368
18.1.8
Bundles of Lie superalgebras
................. 369
18.2
Superconnections
........................... 369
18.2.1
Extending
superconnections
to the bundle of endomor-
phisms
............................. 370
18.2.2
Supercurvature
......................... 370
18.2.3
The tensor product of two
superconnections
........ 371
18.2.4
The exterior components of
a superconnection
....... 371
18.2.5
A local computation
...................... 372
18.3
S
up
er
connections and principal bundles
............... 372
18.3.1
Recalling some definitions
................... 372
18.3.2
Generalizing the above to
superconnections
......... 374
18.4
Clifford Bundles and Clifford
superconnections
........... 375
19
Semi-Riemannian submersions.
377
19.1
Submersions
.............................. 378
19.2
The fundamental tensors of a submersion
.............. 380
19.2.1
The tensor
Τ
.......................... 380
19.2.2
The tensor A
.......................... 381
19.2.3
Covariant derivatives of
Τ
and A
............... 382
19.2.4
The fundamental tensors for a warped product
......
384
19.3
Curvature
................................ 385
19.3.1
Curvature for warped products
................ 390
19.3.2
Sectional curvature
...................... 393
19.4
Reductive homogeneous spaces
.................... 394
19.4.1
Homogeneous spaces
...................... 394
19.4.2
Normal symmetric spaces
................... 394
19.4.3
Orthogonal groups
....................... 396
19.4.4
Dual Grassmannians
..................... 397
19.5 Schwarzschild
as a warped product
.................. 399
Bibliography
................................. 401
Index
..................................... 402
Curvature in
Mathematics and Physics
Shlomo
Sternberg
This original text for courses in differential geometry is geared toward
advanced undergraduate and graduate majors in math and physics.
Based on an advanced class taught by a world-renowned mathematician
for more than fifty years, the treatment introduces semi-Riemannian
geometry and its principal physical application, Einstein s theory of
general relativity, using the Cartan exterior calculus as a principal tool.
Starting with an introduction to the various curvatures associated to
a hypersurface embedded in Euclidean space, the text advances to a
brief review of the differential and integral calculus on manifolds. A
discussion of the fundamental notions of linear connections and their
curvatures follows, along with considerations of Levi-Civita s theorem,
bi-invariant metrics on a Lie group, Cartan calculations, Gauss s
lemma, and variational formulas. Additional topics include the
Hopf-
Rinow, Myer s, and Frobenius theorems; special and general relativity;
connections on principal and associated bundles; the star operator;
superconnections;
and semi-Riemannian submersions. Prerequisites
include linear algebra and advanced calculus, preferably in the language
of differential forms.
Dover
(2012)
original publication.
|
| any_adam_object | 1 |
| author | Sternberg, Shlomo 1936- |
| author_GND | (DE-588)121101762 |
| author_facet | Sternberg, Shlomo 1936- |
| author_role | aut |
| author_sort | Sternberg, Shlomo 1936- |
| author_variant | s s ss |
| building | Verbundindex |
| bvnumber | BV041371361 |
| callnumber-first | Q - Science |
| callnumber-label | QA645 |
| callnumber-raw | QA645 |
| callnumber-search | QA645 |
| callnumber-sort | QA 3645 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 370 |
| classification_tum | PHY 014f MAT 530f |
| ctrlnum | (OCoLC)822619128 (DE-599)GBV730078914 |
| dewey-full | 516.362 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.362 |
| dewey-search | 516.362 |
| dewey-sort | 3516.362 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV041371361 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:55:12Z |
| institution | BVB |
| isbn | 0486478556 9780486478555 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026819508 |
| oclc_num | 822619128 |
| open_access_boolean | |
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| owner_facet | DE-19 DE-BY-UBM DE-739 DE-20 DE-91G DE-BY-TUM DE-706 DE-355 DE-BY-UBR |
| physical | 405 S. Ill., graph. Darst. 24 cm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Dover Publ. |
| record_format | marc |
| spelling | Sternberg, Shlomo 1936- Verfasser (DE-588)121101762 aut Curvature in mathematics and physics Shlomo Sternberg 1. publ. Mineola, NY Dover Publ. 2012 405 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 401 und Index Curvature Semi-Riemannian geometry Geometry, Differential Differential calculus Algebras, Linear Relativity (Physics) Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Krümmung (DE-588)4128765-4 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Krümmung (DE-588)4128765-4 s Mathematische Physik (DE-588)4037952-8 s DE-604 DE-601 pdf/application http://zbmath.org/?q=an:1257.53001 Zentralblatt MATH Inhaltstext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026819508&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026819508&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Sternberg, Shlomo 1936- Curvature in mathematics and physics Curvature Semi-Riemannian geometry Geometry, Differential Differential calculus Algebras, Linear Relativity (Physics) Mathematische Physik (DE-588)4037952-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd Krümmung (DE-588)4128765-4 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4012248-7 (DE-588)4128765-4 |
| title | Curvature in mathematics and physics |
| title_auth | Curvature in mathematics and physics |
| title_exact_search | Curvature in mathematics and physics |
| title_full | Curvature in mathematics and physics Shlomo Sternberg |
| title_fullStr | Curvature in mathematics and physics Shlomo Sternberg |
| title_full_unstemmed | Curvature in mathematics and physics Shlomo Sternberg |
| title_short | Curvature in mathematics and physics |
| title_sort | curvature in mathematics and physics |
| topic | Curvature Semi-Riemannian geometry Geometry, Differential Differential calculus Algebras, Linear Relativity (Physics) Mathematische Physik (DE-588)4037952-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd Krümmung (DE-588)4128765-4 gnd |
| topic_facet | Curvature Semi-Riemannian geometry Geometry, Differential Differential calculus Algebras, Linear Relativity (Physics) Mathematische Physik Differentialgeometrie Krümmung |
| url | http://zbmath.org/?q=an:1257.53001 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026819508&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026819508&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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