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# Some Properties of Differentiable Varieties and Transformations: With Special Reference to the Analytic and Algebraic Cases by Beniamino Segre (Paperback, 2011)

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## About this product

### Key Features

- Author(s)Beniamino Segre
- PublisherSpringer-Verlag Berlin and Heidelberg GmbH & Co. KG
- Date of Publication11/11/2011
- Language(s)English
- FormatPaperback
- ISBN-103642650082
- ISBN-139783642650086
- GenreMathematics
- Series TitleErgebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
- Series Part/Volume Number13

### Publication Data

- Place of PublicationBerlin
- Country of PublicationGermany
- ImprintSpringer-Verlag Berlin and Heidelberg GmbH & Co. K
- Content NoteX, 198 p.

### Dimensions

- Weight320 g
- Width152 mm
- Height229 mm
- Pagination198

### Editorial Details

- Edition Statement2nd ed. 1971. Softcover reprint of the original 2nd ed. 1971

### Description

- Table Of ContentsOne. 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

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