Riemannian geometry during the second half of the twentieth century:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2002
|
| Edition: | Reprinted with corrections |
| Series: | University lecture series
Volume 17 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Erscheint auch in: Jahresbericht der Deutschen Mathematiker-Vereinigung ; 100,2 |
| Physical Description: | xi, 192 Seiten Diagramme |
| ISBN: | 0821820524 9780821820520 |
Staff View
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV021443206 | ||
| 003 | DE-604 | ||
| 005 | 20230330 | ||
| 007 | t | ||
| 008 | 060216s2002 |||| |||| 00||| eng d | ||
| 020 | |a 0821820524 |9 0-8218-2052-4 | ||
| 020 | |a 9780821820520 |9 978-0-8218-2052-0 | ||
| 035 | |a (OCoLC)428776648 | ||
| 035 | |a (DE-599)BVBBV021443206 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 045 | |a x5x9 | ||
| 049 | |a DE-19 |a DE-703 |a DE-11 |a DE-355 |a DE-91G | ||
| 050 | 0 | |a QA649 | |
| 082 | 0 | |a 516.3/73 |2 21 | |
| 084 | |a SG 580 |0 (DE-625)143067: |2 rvk | ||
| 084 | |a SI 165 |0 (DE-625)143085: |2 rvk | ||
| 100 | 1 | |a Berger, Marcel |d 1927-2016 |e Verfasser |0 (DE-588)12047672X |4 aut | |
| 245 | 1 | 0 | |a Riemannian geometry during the second half of the twentieth century |c Marcel Berger |
| 250 | |a Reprinted with corrections | ||
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c 2002 | |
| 264 | 4 | |c © 2000 | |
| 300 | |a xi, 192 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a University lecture series |v Volume 17 | |
| 500 | |a Erscheint auch in: Jahresbericht der Deutschen Mathematiker-Vereinigung ; 100,2 | ||
| 648 | 7 | |a Geschichte 1950-2000 |2 gnd |9 rswk-swf | |
| 648 | 7 | |a Geschichte 1950-1998 |2 gnd |9 rswk-swf | |
| 650 | 7 | |a Differentiaalmeetkunde |2 gtt | |
| 650 | 4 | |a Riemannsche Geometrie - Geschichte | |
| 650 | 4 | |a Geometry, Riemannian | |
| 650 | 0 | 7 | |a Riemannsche Geometrie |0 (DE-588)4128462-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Riemannsche Geometrie |0 (DE-588)4128462-8 |D s |
| 689 | 0 | 1 | |a Geschichte 1950-2000 |A z |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Riemannsche Geometrie |0 (DE-588)4128462-8 |D s |
| 689 | 1 | 1 | |a Geschichte 1950-1998 |A z |
| 689 | 1 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-2215-8 |
| 830 | 0 | |a University lecture series |v Volume 17 |w (DE-604)BV004153846 |9 17 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014660222&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-014660222 | ||
Record in the Search Index
| _version_ | 1804135157477670912 |
|---|---|
| adam_text | Contents
-1.
Introduction
ix
Note
on the second printing
xi
0.
Riemannian Geometry up to
1950 1
A. Gauss, Riemann,
Christoffel
and Levi-Civita
1
B. Van Mangoldt,
Hadamard,
Elie Cartan
and Heinz
Hopf 3
C. Synge,
Myers. Preissmann: The use of geometric tools
4
D. Hodge, harmonic forms and the Bochner technique:
The use of analysis
5
E. Allendoerfer. Weil and Chern
6
F. Existing tools and a brief look at the new ones
8
G. Existing examples and a brief look at the new ones
10
1.
Comments on the Main Topics I. II. HI. IV. V
under Consideration
13
I. Curvature and Topology
17
A. Pinching Problems
17
1.
Introduction
17
2.
Positive Pinching
18
A digression: Comparison theorems
19
3.
Pinching around zero
25
4.
Negative pinching
26
B. Curvature of a given sign
27
1.
The positive side: Sectional curvature
27
2.
The positive side:
Ricci
curvature
33
3.
The positive side: Scalar curvature
35
4.
The negative side: Sectional curvature
38
5.
The negative side:
Ricci
curvature
43
C. Finiteness, compactness, collapsing
and the space of all Riemannian structures
44
1.
Finiteness results
44
A digression: Does the curvature determine the metric?
46
2.
Compactness, convergence results
49
3.
The set of all Riemannian metrics: Collapsing
52
II. The Geometrical Hierarchy of Riemannian Manifolds: Space Forms
57
A. The constant curvature case
57
B. Space forms of rank one
59
C. Space forms of symmetric spaces of rank larger than one
60
viii CONTENTS
III.
The Set of Riemannian Structures on a Given Compact Manifold:
Is There a Best Metric?
63
A. The problem
63
B. The minimal volume and min|
|ß||d/2 64
C.
The case of Einstein manifolds
65
Digression: The Yamabe Problem
65
D. Some topological closures
69
E. The fractal nature of TIS(M) according to Nabutovsky
70
IV. The Spectrum, the Eigenfunctions
75
V. Periodic Geodesies, the Geodesic Flow
85
A. Periodic geodesies
85
A digression: Geodesies joining two points
87
B. The geodesic flow (geometry and dynamics)
92
Digression: Entropies in Riemannian Geometry
96
TOP. Some Other Riemannian Geometry Topics of Interest
99
1.
Volumes
99
A. Bishop s Theorem
99
B. The isoperimetric profile
99
С
The
embolie
volume
102
D. The minimal volume
102
E. The systolic story
104
2.
Isometric embedding
107
3.
Holonomy groups and special metrics: Another (very restricted)
Riemannian hierarchy,
Kahler
manifolds
108
A. Holonomy groups
108
B.
Kahler
manifolds
110
4.
Cut-loci
112
5.
Noncompact manifolds
114
6.
Bundles over Riemannian manifolds
117
A. Exterior differential forms (and some others)
117
B. Spinors
120
C. Various other bundles
123
7.
Harmonic maps between Riemannian manifolds
125
8.
Low dimensional Riemannian geometry
126
9.
Some generalizations of Riemannian geometry
127
10.
Submanifolds
133
A. The case of surfaces in R3
134
B. Higher dimensions
135
С
Geometric measure theory and pseudo-holomorphic curves
136
Bibliography
137
Additional Bibliography
171
Subject Index
175
Author Index
187
|
| adam_txt |
Contents
-1.
Introduction
ix
Note
on the second printing
xi
0.
Riemannian Geometry up to
1950 1
A. Gauss, Riemann,
Christoffel
and Levi-Civita
1
B. Van Mangoldt,
Hadamard,
Elie Cartan
and Heinz
Hopf 3
C. Synge,
Myers. Preissmann: The use of geometric tools
4
D. Hodge, harmonic forms and the Bochner technique:
The use of analysis
5
E. Allendoerfer. Weil and Chern
6
F. Existing tools and a brief look at the new ones
8
G. Existing examples and a brief look at the new ones
10
1.
Comments on the Main Topics I. II. HI. IV. V
under Consideration
13
I. Curvature and Topology
17
A. Pinching Problems
17
1.
Introduction
17
2.
Positive Pinching
18
A digression: Comparison theorems
19
3.
Pinching around zero
25
4.
Negative pinching
26
B. Curvature of a given sign
27
1.
The positive side: Sectional curvature
27
2.
The positive side:
Ricci
curvature
33
3.
The positive side: Scalar curvature
35
4.
The negative side: Sectional curvature
38
5.
The negative side:
Ricci
curvature
43
C. Finiteness, compactness, collapsing
and the space of all Riemannian structures
44
1.
Finiteness results
44
A digression: Does the curvature determine the metric?
46
2.
Compactness, convergence results
49
3.
The set of all Riemannian metrics: Collapsing
52
II. The Geometrical Hierarchy of Riemannian Manifolds: Space Forms
57
A. The constant curvature case
57
B. Space forms of rank one
59
C. Space forms of symmetric spaces of rank larger than one
60
viii CONTENTS
III.
The Set of Riemannian Structures on a Given Compact Manifold:
Is There a Best Metric?
63
A. The problem
63
B. The minimal volume and min|
|ß||d/2 64
C.
The case of Einstein manifolds
65
Digression: The Yamabe Problem
65
D. Some topological closures
69
E. The fractal nature of TIS(M) according to Nabutovsky
70
IV. The Spectrum, the Eigenfunctions
75
V. Periodic Geodesies, the Geodesic Flow
85
A. Periodic geodesies
85
A digression: Geodesies joining two points
87
B. The geodesic flow (geometry and dynamics)
92
Digression: Entropies in Riemannian Geometry
96
TOP. Some Other Riemannian Geometry Topics of Interest
99
1.
Volumes
99
A. Bishop's Theorem
99
B. The isoperimetric profile
99
С
The
embolie
volume
102
D. The minimal volume
102
E. The systolic story
104
2.
Isometric embedding
107
3.
Holonomy groups and special metrics: Another (very restricted)
Riemannian hierarchy,
Kahler
manifolds
108
A. Holonomy groups
108
B.
Kahler
manifolds
110
4.
Cut-loci
112
5.
Noncompact manifolds
114
6.
Bundles over Riemannian manifolds
117
A. Exterior differential forms (and some others)
117
B. Spinors
120
C. Various other bundles
123
7.
Harmonic maps between Riemannian manifolds
125
8.
Low dimensional Riemannian geometry
126
9.
Some generalizations of Riemannian geometry
127
10.
Submanifolds
133
A. The case of surfaces in R3
134
B. Higher dimensions
135
С
Geometric measure theory and pseudo-holomorphic curves
136
Bibliography
137
Additional Bibliography
171
Subject Index
175
Author Index
187 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Berger, Marcel 1927-2016 |
| author_GND | (DE-588)12047672X |
| author_facet | Berger, Marcel 1927-2016 |
| author_role | aut |
| author_sort | Berger, Marcel 1927-2016 |
| author_variant | m b mb |
| building | Verbundindex |
| bvnumber | BV021443206 |
| callnumber-first | Q - Science |
| callnumber-label | QA649 |
| callnumber-raw | QA649 |
| callnumber-search | QA649 |
| callnumber-sort | QA 3649 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SG 580 SI 165 |
| ctrlnum | (OCoLC)428776648 (DE-599)BVBBV021443206 |
| dewey-full | 516.3/73 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/73 |
| dewey-search | 516.3/73 |
| dewey-sort | 3516.3 273 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | Reprinted with corrections |
| era | Geschichte 1950-2000 gnd Geschichte 1950-1998 gnd |
| era_facet | Geschichte 1950-2000 Geschichte 1950-1998 |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02210nam a2200553 cb4500</leader><controlfield tag="001">BV021443206</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20230330 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">060216s2002 |||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0821820524</subfield><subfield code="9">0-8218-2052-4</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780821820520</subfield><subfield code="9">978-0-8218-2052-0</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)428776648</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV021443206</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="045" ind1=" " ind2=" "><subfield code="a">x5x9</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-19</subfield><subfield code="a">DE-703</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-91G</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA649</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">516.3/73</subfield><subfield code="2">21</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SG 580</subfield><subfield code="0">(DE-625)143067:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SI 165</subfield><subfield code="0">(DE-625)143085:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Berger, Marcel</subfield><subfield code="d">1927-2016</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)12047672X</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Riemannian geometry during the second half of the twentieth century</subfield><subfield code="c">Marcel Berger</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">Reprinted with corrections</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Providence, Rhode Island</subfield><subfield code="b">American Mathematical Society</subfield><subfield code="c">2002</subfield></datafield><datafield tag="264" ind1=" " ind2="4"><subfield code="c">© 2000</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">xi, 192 Seiten</subfield><subfield code="b">Diagramme</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">University lecture series</subfield><subfield code="v">Volume 17</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Erscheint auch in: Jahresbericht der Deutschen Mathematiker-Vereinigung ; 100,2</subfield></datafield><datafield tag="648" ind1=" " ind2="7"><subfield code="a">Geschichte 1950-2000</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="648" ind1=" " ind2="7"><subfield code="a">Geschichte 1950-1998</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Differentiaalmeetkunde</subfield><subfield code="2">gtt</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Riemannsche Geometrie - Geschichte</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Geometry, Riemannian</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Riemannsche Geometrie</subfield><subfield code="0">(DE-588)4128462-8</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Riemannsche Geometrie</subfield><subfield code="0">(DE-588)4128462-8</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Geschichte 1950-2000</subfield><subfield code="A">z</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="689" ind1="1" ind2="0"><subfield code="a">Riemannsche Geometrie</subfield><subfield code="0">(DE-588)4128462-8</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2="1"><subfield code="a">Geschichte 1950-1998</subfield><subfield code="A">z</subfield></datafield><datafield tag="689" ind1="1" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Online-Ausgabe</subfield><subfield code="z">978-1-4704-2215-8</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">University lecture series</subfield><subfield code="v">Volume 17</subfield><subfield code="w">(DE-604)BV004153846</subfield><subfield code="9">17</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Regensburg</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014660222&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-014660222</subfield></datafield></record></collection> |
| id | DE-604.BV021443206 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T14:03:45Z |
| indexdate | 2024-07-09T20:36:02Z |
| institution | BVB |
| isbn | 0821820524 9780821820520 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014660222 |
| oclc_num | 428776648 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-703 DE-11 DE-355 DE-BY-UBR DE-91G DE-BY-TUM |
| owner_facet | DE-19 DE-BY-UBM DE-703 DE-11 DE-355 DE-BY-UBR DE-91G DE-BY-TUM |
| physical | xi, 192 Seiten Diagramme |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | University lecture series |
| series2 | University lecture series |
| spelling | Berger, Marcel 1927-2016 Verfasser (DE-588)12047672X aut Riemannian geometry during the second half of the twentieth century Marcel Berger Reprinted with corrections Providence, Rhode Island American Mathematical Society 2002 © 2000 xi, 192 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier University lecture series Volume 17 Erscheint auch in: Jahresbericht der Deutschen Mathematiker-Vereinigung ; 100,2 Geschichte 1950-2000 gnd rswk-swf Geschichte 1950-1998 gnd rswk-swf Differentiaalmeetkunde gtt Riemannsche Geometrie - Geschichte Geometry, Riemannian Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 s Geschichte 1950-2000 z DE-604 Geschichte 1950-1998 z Erscheint auch als Online-Ausgabe 978-1-4704-2215-8 University lecture series Volume 17 (DE-604)BV004153846 17 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014660222&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Berger, Marcel 1927-2016 Riemannian geometry during the second half of the twentieth century University lecture series Differentiaalmeetkunde gtt Riemannsche Geometrie - Geschichte Geometry, Riemannian Riemannsche Geometrie (DE-588)4128462-8 gnd |
| subject_GND | (DE-588)4128462-8 |
| title | Riemannian geometry during the second half of the twentieth century |
| title_auth | Riemannian geometry during the second half of the twentieth century |
| title_exact_search | Riemannian geometry during the second half of the twentieth century |
| title_exact_search_txtP | Riemannian geometry during the second half of the twentieth century |
| title_full | Riemannian geometry during the second half of the twentieth century Marcel Berger |
| title_fullStr | Riemannian geometry during the second half of the twentieth century Marcel Berger |
| title_full_unstemmed | Riemannian geometry during the second half of the twentieth century Marcel Berger |
| title_short | Riemannian geometry during the second half of the twentieth century |
| title_sort | riemannian geometry during the second half of the twentieth century |
| topic | Differentiaalmeetkunde gtt Riemannsche Geometrie - Geschichte Geometry, Riemannian Riemannsche Geometrie (DE-588)4128462-8 gnd |
| topic_facet | Differentiaalmeetkunde Riemannsche Geometrie - Geschichte Geometry, Riemannian Riemannsche Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014660222&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004153846 |
| work_keys_str_mv | AT bergermarcel riemanniangeometryduringthesecondhalfofthetwentiethcentury |