Some properties of differentiable varieties and transformations: with special reference to the analytic and algebraic cases
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1971
|
| Edition: | 2. ed. |
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete
13 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 195 S. |
Staff View
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| 245 | 1 | 0 | |a Some properties of differentiable varieties and transformations |b with special reference to the analytic and algebraic cases |
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| 650 | 4 | |a Geometry, Algebraic | |
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Record in the Search Index
| _version_ | 1804136130879160320 |
|---|---|
| adam_text | Contents
Part One. Differential Invariants of Point and Dual Transformations .... 1
§ 1. Local metrical study of point transformations . 1
§ 2. Some topologico differential invariants 3
§ 3. Protective construction of the above invariants 5
§ 4. Local metrical study of the dual transformations 7
§ 5. Calculation of the first order differential invariants just considered . 9
§ 6. Some particular transformations. Relations between densities ... 11
§ 7. The curvature of hypersurfaces and of Pfaffian forms 12
Historical Notes and Bibliography 14
Part Two. Local Properties of Analytic Transformations at their United Points 14
§ 8. Coefficients of dilatation and residues of transformations in the
analytic field 14
§ 9. Transfer to the Riemann variety 16
§ 10. Formal changes of coordinates 17
§ 11. Formal reduction to the canonical form for the arithmetically general
transformations 19
§ 12. The case of arithmetically special transformations 21
§ 13. Criteria of convergence for the reduction procedure in the general case 22
§ 14. Iteration and permutability of analytic transformations 26
§ 15. On the united points of cyclic transformations 29
§ 16. Arithmetically general transformations not representable linearly . . 31
Historical Notes and Bibliography 34
Part Three. Invariants of Contact and of Osculation. The Concept of Cross ratio
in Differential Geometry 35
§ 17. Projective invariants of two curves having the same osculating spaces
at a point 36
§ 18. A notable metric case 38
§ 19. An important extension . 39
§ 20. Projective invariants of contact of differential elements of any
dimension 41
§ 21. Two applications 43
§ 22. On certain varieties generated by quadrics 44
§ 23. The notion of cross ratio on certain surfaces 46
§ 24. Applications to various branches of differential geometry 48
§ 25. Some extensions 50
Historical Notes and Bibliography 52
Part Four. Principal and Projective Curves of a Surface, and Some Applications 53
§ 26. Some results of projective differential geometry 53
§ 27. The definition and main properties of the principal and projective
curves 55
§ 28. Further properties of the above curves 57
§ 29. The use of the Laplace invariants and of the infinitesimal invariants 59
§ 30. Some classes of surfaces on which the concept of cross ratio is parti¬
cularly simple 61
VIII Contents
§ 31. Point correspondences which conserve the projective curves .... 64
§ 32. Point correspondences which preserve the principal lines 66
§ 33. On the plane cone curves of a surface 68
Historical Notes and Bibliography 69
Part Five. Some Differential Properties in the Large of Algebraic Curves, their
Intersections, and Self correspondences 69
§ 34. The residues of correspondences on curves, and a topological invariant
of intersection of two curves on a surface which contains two privi¬
leged pencils of curves 69
§ 35. A complement of the correspondence principle on algebraic curves. . 73
§ 36. A geometric characterization of Abelian integrals and their residues 76
§ 37. The first applications 79
§ 38. The equation of Jacobi, and some consequences 81
§ 39. The relation of Reiss, and some extensions 83
§ 40. Further algebra differential properties 86
Historical Notes and Bibliography 88
Part Six. Extensions to Algebraic Varieties 88
§ 41. Generalizations of the equation of Jacobi 89
§ 42. Generalizations of the relation of Reiss 91
§ 43. The residue of an analytic transformation at a simple united point 93
§ 44. Some important particular cases 95
§ 45. Relations between residues at the same point 97
§ 46. The total residues of correspondences of valency zero on algebraic
varieties 98
§ 47. The residues at isolated united points with arbitrary multiplicities 100
§ 48. Extensions to algebraic correspondences of arbitrary valency . . . .103
§ 49. Applications to algebraic correspondences of a projective space into
itself 105
Historical Notes and Bibliography 107
Part Seven. Veronese Varieties and Modules of Algebraic Forms 108
§ 50. M regular points of differentiable varieties 108
§ 51. Some special properties of n regular points of differentiable varieties 110
§ 52. On the freedom of hypersurfaces having assigned multiplicities at a set
of points 114
§ 53. On the effective dimension of certain linear systems of hypersurfaces 115
§ 54. Two relations of Lasker concerning modules of hypersurfaces . . . 117
§ 55. Some important criteria for a hypersurface to belong to a given module 119
§ 56. Some properties of the osculating spaces at the points of a Veronese
variety Vf 121
§ 57. The ambients of certain subvarieties of V1^ 123
§ 58. The isolated multiple intersections of d primals on Vff 125
§ 59. The regular multiple intersections on V^ 126
§ 60. A special property of the space associated with an isolated intersection
on F j in the simple case 128
§ 61. On a theorem of Torelli and some complements 130
Historical Notes and Bibliography 131
Part Eight. Linear Partial Differential Equations 132
§ 62. Preliminary observations 132
§ 63. The reduction of differential equations to a canonical form 133
§ 64. Remarks on the solution of the differential equations 134
§ 65. The construction of the conditions of integrability 136
Contents IX
§ 66. The conditions of compatibility for a system of linear partial differential
equations in one unknown 137
§ 67. The analytic case where the characteristic hypersurfaces intersect
regularly 139
§ 68. An extension to the non analytic case 142
§ 69. Some remarks on sets of linear partial differential equations in several
unknowns 144
§ 70. The solution of a system of homogeneous equations 146
§ 71. The resolving system associated with a general set of m differential
equations in m unknowns 149
Historical Notes and Bibliography 151
Part Nine. Projective Differential Geometry of Systems of Linear Partial Dif¬
ferential Equations 152
§ 72. y osculating spaces to a variety 152
§ 73. Surfaces representing Laplace equations 153
§ 74. The hyperbolic case 153
§ 75. The parabolic case 157
§ 76. Surfaces representing differential equations of arbitrary order . . 158
§ 77. Varieties of arbitrary dimension representing Laplace equations 158
§ 78. Generalized developables 159
§ 79. Varieties of arbitrary order representing differential equations of
arbitrary order 160
§ 80. The postulation of varieties by conditions on their y osculating
spaces 162
Historical Notes and Bibliography 163
Part Ten. Correspondences between Topological Varieties 164
§ 81. Products of topological varieties 164
§ 82. Correspondences and relations 165
§ 83. Inverse correspondences 166
§ 84. Homologous correspondences 167
§ 85. Topological invariants of correspondences between topological varieties 168
§ 86. Arithmetic and algebraic invariants 169
§ 87. Geometric invariants 170
§ 88. t correspondences on topological varieties 171
§ 89. Semiregular correspondences and their products 173
§ 90. Characteristic integers of a semi regular correspondence 175
§ 91. Involutory elementary ^ correspondences 175
§ 92. Algebraic and skew algebraic involutory transformations 177
§ 93. An extension of Zeuthen s formula to the topological domain . . . 178
§ 94. One valued elementary correspondences 180
§ 95. Correspondences represented by differentiate varieties 181
Historical Notes and Bibliography 183
Bibliography 184
Author Index 191
Analytic Index 193
|
| adam_txt |
Contents
Part One. Differential Invariants of Point and Dual Transformations . 1
§ 1. Local metrical study of point transformations . 1
§ 2. Some topologico differential invariants 3
§ 3. Protective construction of the above invariants 5
§ 4. Local metrical study of the dual transformations 7
§ 5. Calculation of the first order differential invariants just considered . 9
§ 6. Some particular transformations. Relations between densities . 11
§ 7. The curvature of hypersurfaces and of Pfaffian forms 12
Historical Notes and Bibliography 14
Part Two. Local Properties of Analytic Transformations at their United Points 14
§ 8. Coefficients of dilatation and residues of transformations in the
analytic field 14
§ 9. Transfer to the Riemann variety 16
§ 10. Formal changes of coordinates 17
§ 11. Formal reduction to the canonical form for the arithmetically general
transformations 19
§ 12. The case of arithmetically special transformations 21
§ 13. Criteria of convergence for the reduction procedure in the general case 22
§ 14. Iteration and permutability of analytic transformations 26
§ 15. On the united points of cyclic transformations 29
§ 16. Arithmetically general transformations not representable linearly . . 31
Historical Notes and Bibliography 34
Part Three. Invariants of Contact and of Osculation. The Concept of Cross ratio
in Differential Geometry 35
§ 17. Projective invariants of two curves having the same osculating spaces
at a point 36
§ 18. A notable metric case 38
§ 19. An important extension . 39
§ 20. Projective invariants of contact of differential elements of any
dimension 41
§ 21. Two applications 43
§ 22. On certain varieties generated by quadrics 44
§ 23. The notion of cross ratio on certain surfaces 46
§ 24. Applications to various branches of differential geometry 48
§ 25. Some extensions 50
Historical Notes and Bibliography 52
Part Four. Principal and Projective Curves of a Surface, and Some Applications 53
§ 26. Some results of projective differential geometry 53
§ 27. The definition and main properties of the principal and projective
curves 55
§ 28. Further properties of the above curves 57
§ 29. The use of the Laplace invariants and of the infinitesimal invariants 59
§ 30. Some classes of surfaces on which the concept of cross ratio is parti¬
cularly simple 61
VIII Contents
§ 31. Point correspondences which conserve the projective curves . 64
§ 32. Point correspondences which preserve the principal lines 66
§ 33. On the plane cone curves of a surface 68
Historical Notes and Bibliography 69
Part Five. Some Differential Properties in the Large of Algebraic Curves, their
Intersections, and Self correspondences 69
§ 34. The residues of correspondences on curves, and a topological invariant
of intersection of two curves on a surface which contains two privi¬
leged pencils of curves 69
§ 35. A complement of the correspondence principle on algebraic curves. . 73
§ 36. A geometric characterization of Abelian integrals and their residues 76
§ 37. The first applications 79
§ 38. The equation of Jacobi, and some consequences 81
§ 39. The relation of Reiss, and some extensions 83
§ 40. Further algebra differential properties 86
Historical Notes and Bibliography 88
Part Six. Extensions to Algebraic Varieties 88
§ 41. Generalizations of the equation of Jacobi 89
§ 42. Generalizations of the relation of Reiss 91
§ 43. The residue of an analytic transformation at a simple united point 93
§ 44. Some important particular cases 95
§ 45. Relations between residues at the same point 97
§ 46. The total residues of correspondences of valency zero on algebraic
varieties 98
§ 47. The residues at isolated united points with arbitrary multiplicities 100
§ 48. Extensions to algebraic correspondences of arbitrary valency . . . .103
§ 49. Applications to algebraic correspondences of a projective space into
itself 105
Historical Notes and Bibliography 107
Part Seven. Veronese Varieties and Modules of Algebraic Forms 108
§ 50. M regular points of differentiable varieties 108
§ 51. Some special properties of n regular points of differentiable varieties 110
§ 52. On the freedom of hypersurfaces having assigned multiplicities at a set
of points 114
§ 53. On the effective dimension of certain linear systems of hypersurfaces 115
§ 54. Two relations of Lasker concerning modules of hypersurfaces . . . 117
§ 55. Some important criteria for a hypersurface to belong to a given module 119
§ 56. Some properties of the osculating spaces at the points of a Veronese
variety Vf 121
§ 57. The ambients of certain subvarieties of V1^ 123
§ 58. The isolated multiple intersections of d primals on Vff 125
§ 59. The regular multiple intersections on V^ 126
§ 60. A special property of the space associated with an isolated intersection
on F j in the simple case 128
§ 61. On a theorem of Torelli and some complements 130
Historical Notes and Bibliography 131
Part Eight. Linear Partial Differential Equations 132
§ 62. Preliminary observations 132
§ 63. The reduction of differential equations to a canonical form 133
§ 64. Remarks on the solution of the differential equations 134
§ 65. The construction of the conditions of integrability 136
Contents IX
§ 66. The conditions of compatibility for a system of linear partial differential
equations in one unknown 137
§ 67. The analytic case where the characteristic hypersurfaces intersect
regularly 139
§ 68. An extension to the non analytic case 142
§ 69. Some remarks on sets of linear partial differential equations in several
unknowns 144
§ 70. The solution of a system of homogeneous equations 146
§ 71. The resolving system associated with a general set of m differential
equations in m unknowns 149
Historical Notes and Bibliography 151
Part Nine. Projective Differential Geometry of Systems of Linear Partial Dif¬
ferential Equations 152
§ 72. y osculating spaces to a variety 152
§ 73. Surfaces representing Laplace equations 153
§ 74. The hyperbolic case 153
§ 75. The parabolic case 157
§ 76. Surfaces representing differential equations of arbitrary order . . 158
§ 77. Varieties of arbitrary dimension representing Laplace equations 158
§ 78. Generalized developables 159
§ 79. Varieties of arbitrary order representing differential equations of
arbitrary order 160
§ 80. The postulation of varieties by conditions on their y osculating
spaces 162
Historical Notes and Bibliography 163
Part Ten. Correspondences between Topological Varieties 164
§ 81. Products of topological varieties 164
§ 82. Correspondences and relations 165
§ 83. Inverse correspondences 166
§ 84. Homologous correspondences 167
§ 85. Topological invariants of correspondences between topological varieties 168
§ 86. Arithmetic and algebraic invariants 169
§ 87. Geometric invariants 170
§ 88. t correspondences on topological varieties 171
§ 89. Semiregular correspondences and their products 173
§ 90. Characteristic integers of a semi regular correspondence 175
§ 91. Involutory elementary ^ correspondences 175
§ 92. Algebraic and skew algebraic involutory transformations 177
§ 93. An extension of Zeuthen's formula to the topological domain . . . 178
§ 94. One valued elementary correspondences 180
§ 95. Correspondences represented by differentiate varieties 181
Historical Notes and Bibliography 183
Bibliography 184
Author Index 191
Analytic Index 193 |
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| id | DE-604.BV022156456 |
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| indexdate | 2024-07-09T20:51:31Z |
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| physical | IX, 195 S. |
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| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete |
| spelling | Segre, Beniamino Verfasser aut Some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases 2. ed. Berlin [u.a.] Springer 1971 IX, 195 S. txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete 13 Géométrie algébrique ram Transformations (mathématiques) ram Differential invariants Geometry, Algebraic Transformations (Mathematics) Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd rswk-swf Projektive Varietät (DE-588)4327070-0 gnd rswk-swf Transformation Mathematik (DE-588)4060637-5 gnd rswk-swf Projektive Varietät (DE-588)4327070-0 s DE-604 Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 s Transformation Mathematik (DE-588)4060637-5 s 1\p DE-604 Ergebnisse der Mathematik und ihrer Grenzgebiete 13 (DE-604)BV005871160 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015371117&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Segre, Beniamino Some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases Ergebnisse der Mathematik und ihrer Grenzgebiete Géométrie algébrique ram Transformations (mathématiques) ram Differential invariants Geometry, Algebraic Transformations (Mathematics) Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Projektive Varietät (DE-588)4327070-0 gnd Transformation Mathematik (DE-588)4060637-5 gnd |
| subject_GND | (DE-588)4012269-4 (DE-588)4327070-0 (DE-588)4060637-5 |
| title | Some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases |
| title_auth | Some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases |
| title_exact_search | Some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases |
| title_exact_search_txtP | Some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases |
| title_full | Some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases |
| title_fullStr | Some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases |
| title_full_unstemmed | Some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases |
| title_short | Some properties of differentiable varieties and transformations |
| title_sort | some properties of differentiable varieties and transformations with special reference to the analytic and algebraic cases |
| title_sub | with special reference to the analytic and algebraic cases |
| topic | Géométrie algébrique ram Transformations (mathématiques) ram Differential invariants Geometry, Algebraic Transformations (Mathematics) Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Projektive Varietät (DE-588)4327070-0 gnd Transformation Mathematik (DE-588)4060637-5 gnd |
| topic_facet | Géométrie algébrique Transformations (mathématiques) Differential invariants Geometry, Algebraic Transformations (Mathematics) Differenzierbare Mannigfaltigkeit Projektive Varietät Transformation Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015371117&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV005871160 |
| work_keys_str_mv | AT segrebeniamino somepropertiesofdifferentiablevarietiesandtransformationswithspecialreferencetotheanalyticandalgebraiccases |