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Some properties of differentiable varieties and transformations; with special reference to the analytic and algebraic cases.

Author: Beniamino Segre; J W P Hirschfeld
Publisher: Berlin, New York, Springer-Verlag, 1971.
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 13.
Edition/Format:   Print book : English : 2d edView all editions and formats
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Additional Physical Format: Online version:
Segre, Beniamino, 1903-
Some properties of differentiable varieties and transformations.
Berlin, New York, Springer-Verlag, 1971
(OCoLC)682108162
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Beniamino Segre; J W P Hirschfeld
ISBN: 354005085X 9783540050858 038705085X 9780387050850
OCLC Number: 138120
Description: ix, 195 pages 24 cm.
Contents: One. Differential Invariants of Point and Dual Transformations.- 1. Local metrical study of point transformations.- 2. Some topologico-differential invariants.- 3. Projective construction of the above invariants.- 4. Local metrical study of the dual transformations.- 5. Calculation of the first order differential invariants just considered.- 6. Some particular transformations. Relations between densities.- 7. The curvature of hypersurfaces and of Pfaffian forms.- Historical Notes and Bibliography.- Two. Local Properties of Analytic Transformations at their United Points.- 8. Coefficients of dilatation and residues of transformations in the analytic field.- 9. Transfer to the Riemann variety.- 10. Formal changes of coordinates.- 11. Formal reduction to the canonical form for the arithmetically general transformations.- 12. The case of arithmetically special transformations.- 13. Criteria of convergence for the reduction procedure in the general case.- 14. Iteration and permutability of analytic transformations.- 15. On the united points of cyclic transformations.- 16. Arithmetically general transformations not representable linearly.- Historical Notes and Bibliography.- Three. Invariants of Contact and of Osculation. The Concept of Cross-ratio in Differential Geometry.- 17. Projective invariants of two curves having the same osculating spaces at a point.- 18. A notable metric case.- 19. An important extension.- 20. Projective invariants of contact of differential elements of any dimension.- 21. Two applications.- 22. On certain varieties generated by quadrics.- 23. The notion of cross-ratio on certain surfaces.- 24. Applications to various branches of differential geometry.- 25. Some extensions.- Historical Notes and Bibliography.- Four. Principal and Projective Curves of a Surface, and Some Applications.- 26. Some results of projective-differential geometry.- 27. The definition and main properties of the principal and projective curves.- 28. Further properties of the above curves.- 29. The use of the Laplace invariants and of the infinitesimal invariants.- 30. Some classes of surfaces on which the concept of cross-ratio is particularly simple.- 31. Point correspondences which conserve the projective curves.- 32. Point correspondences which preserve the principal lines.- 33. On the plane cone curves of a surface.- Historical Notes and Bibliography.- Five. Some Differential Properties in the Large of Algebraic Curves, their Intersections, and Self-correspondences.- 34. The residues of correspondences on curves, and a topological invariant of intersection of two curves on a surface which contains two privileged pencils of curves.- 35. A complement of the correspondence principle on algebraic curves.- 36. A geometric characterization of Abelian integrals and their residues.- 37. The first applications.- 38. The equation of Jacobi, and some consequences.- 39. The relation of Reiss, and some extensions.- 40. Further algebro-differential properties.- Historical Notes and Bibliography.- Six. Extensions to Algebraic Varieties.- 41. Generalizations of the equation of Jacobi.- 42. Generalizations of the relation of Reiss.- 43. The residue of an analytic transformation at a simple united point.- 44. Some important particular cases.- 45. Relations between residues at the same point.- 46. The total residues of correspondences of valency zero on algebraic varieties.- 47. The residues at isolated united points with arbitrary multiplicities.- 48. Extensions to algebraic correspondences of arbitrary valency.- 49. Applications to algebraic correspondences of a projective space into itself.- Historical Notes and Bibliography.- Seven. Veronese Varieties and Modules of Algebraic Forms.- 50. n-regular points of differentiable varieties.- 51. Some special properties of n-regular points of differentiable varieties.- 52. On the freedom of hypersurfaces having assigned multiplicities at a set of points.- 53. On the effective dimension of certain linear systems of hypersurfaces.- 54. Two relations of Lasker concerning modules of hypersurfaces.- 55. Some important criteria for a hypersurface to belong to a given module.- 56. Some properties of the osculating spaces at the points of a Veronese variety Vd(n).- 57. The ambients of certain subvarieties of Vd(n).- 58. The isolated multiple intersections of d primals on Vd(n).- 59. The regular multiple intersections on Vd(n).- 60. A special property of the space associated with an isolated intersection on Vd(n) in the simple case.- 61. On a theorem of Torelli and some complements.- Historical Notes and Bibliography.- Eight. Linear Partial Differential Equations.- 62. Preliminary observations.- 63. The reduction of differential equations to a canonical form.- 64. Remarks on the solution of the differential equations.- 65. The construction of the conditions of integrability.- 66. The conditions of compatibility for a system of linear partial differential equations in one unknown.- 67. The analytic case where the characteristic hypersurfaces intersect regularly.- 68. An extension to the non-analytic case.- 69. Some remarks on sets of linear partial differential equations in several unknowns.- 70. The solution of a system of homogeneous equations.- 71. The resolving system associated with a general set of m differential equations in m unknowns.- Historical Notes and Bibliography.- Nine. Projective Differential Geometry of Systems of Linear Partial Differential Equations.- 72. r-osculating spaces to a variety.- 73. Surfaces representing Laplace equations.- 74. The hyperbolic case.- 75. The parabolic case.- 76. Surfaces representing differential equations of arbitrary order.- 77. Varieties of arbitrary dimension representing Laplace equations.- 78. Generalized developables.- 79. Varieties of arbitrary order representing differential equations of arbitrary order.- 80. The postulation of varieties by conditions on their r-osculating spaces.- Historical Notes and Bibliography.- Ten. Correspondences between Topological Varieties.- 81. Products of topological varieties.- 82. Correspondences and relations.- 83. Inverse correspondences.- 84. Homologous correspondences.- 85. Topological invariants of correspondences between topological varieties.- 86. Arithmetic and algebraic invariants.- 87. Geometric invariants.- 88. i-correspondences on topological varieties.- 89. Semiregular correspondences and their products.- 90. Characteristic integers of a semi-regular correspondence.- 91. Involutory elementary s-correspondences.- 92. Algebraic and skew-algebraic involutory transformations.- 93. An extension of Zeuthen's formula to the topological domain.- 94. One-valued elementary correspondences.- 95. Correspondences represented by differentiable varieties.- Historical Notes and Bibliography.- Author Index.- Analytic Index.
Series Title: Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 13.

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