Differential geometrical relations for a class of formal series

Alexandr Baranovitch

Banach Center Publications (1997)

  • Volume: 40, Issue: 1, page 279-287
  • ISSN: 0137-6934

Abstract

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An extension of the category of local manifolds is considered. Instead of smooth mappings of neighbourhoods of linear spaces as morphisms we deal with formal operator power series (or formal maps). Analogues of the objects appearing on smooth manifolds and vector bundles (vector fields, sections of a bundle, exterior forms, the de Rham complex, connection, etc.) are considered in this way. All the examinations are carried out in algebraic language, for we do not care about the convergence of formal maps. It may be useful for the investigation of some nonlinear differential equations.

How to cite

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Baranovitch, Alexandr. "Differential geometrical relations for a class of formal series." Banach Center Publications 40.1 (1997): 279-287. <http://eudml.org/doc/252256>.

@article{Baranovitch1997,
abstract = {An extension of the category of local manifolds is considered. Instead of smooth mappings of neighbourhoods of linear spaces as morphisms we deal with formal operator power series (or formal maps). Analogues of the objects appearing on smooth manifolds and vector bundles (vector fields, sections of a bundle, exterior forms, the de Rham complex, connection, etc.) are considered in this way. All the examinations are carried out in algebraic language, for we do not care about the convergence of formal maps. It may be useful for the investigation of some nonlinear differential equations.},
author = {Baranovitch, Alexandr},
journal = {Banach Center Publications},
keywords = {local differential geometry; nonlinear differential equations; formal differential geometry},
language = {eng},
number = {1},
pages = {279-287},
title = {Differential geometrical relations for a class of formal series},
url = {http://eudml.org/doc/252256},
volume = {40},
year = {1997},
}

TY - JOUR
AU - Baranovitch, Alexandr
TI - Differential geometrical relations for a class of formal series
JO - Banach Center Publications
PY - 1997
VL - 40
IS - 1
SP - 279
EP - 287
AB - An extension of the category of local manifolds is considered. Instead of smooth mappings of neighbourhoods of linear spaces as morphisms we deal with formal operator power series (or formal maps). Analogues of the objects appearing on smooth manifolds and vector bundles (vector fields, sections of a bundle, exterior forms, the de Rham complex, connection, etc.) are considered in this way. All the examinations are carried out in algebraic language, for we do not care about the convergence of formal maps. It may be useful for the investigation of some nonlinear differential equations.
LA - eng
KW - local differential geometry; nonlinear differential equations; formal differential geometry
UR - http://eudml.org/doc/252256
ER -

References

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  1. [1] M. N. Araslanov, Yu. L. Daletskiǐ, Composition Logarithm in the Class of Formal Operator Power Series, Funct. Anal. Appl. 26 (1992), 57-60. 
  2. [2] A. M. Baranovitch, Yu. L. Daletskiǐ, Differential-Geometric Relations for Formal Operator Power Series Class, Preprint. 
  3. [3] Yu. L. Daletskiǐ, Algebra of Compositions and Non-Linear Equations, appear. 
  4. [4] Yu. L. Daletskiǐ, S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Space, Kluwer Acad. Publ., Dordrecht/Boston/London, 1991. 
  5. [5] I. M. Gel'fand, Yu. L. Daletskiǐ, B. Tsygan, On a Variant of Non-Commutative Differential Geometry, Doklady Academii Nauk USSR 308 (1989), 422-425. 
  6. [6] M. Gerstenhaber, The Cohomology Structure of an Associative Ring, Ann. Math 78 (1963), 59-103. Zbl0131.27302
  7. [7] C. Godbillon, Géométrie Différentielle et Mécanique Analytique, Hermann, Paris, 1969. 

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