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Mat. Sb., 2008, Volume 199, Number 10, Pages 63–86 (Mi msb3935)  

This article is cited in 4 scientific papers (total in 4 papers)

Natural differential operations on manifolds: an algebraic approach

P. I. Katsyloa, D. A. Timashevb

a Scientific Research Institute for System Studies of RAS
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Natural algebraic differential operations on geometric quantities on smooth manifolds are considered. A method for the investigation and classification of such operations is described, the method of IT-reduction. With it the investigation of natural operations reduces to the analysis of rational maps between $k$-jet spaces, which are equivariant with respect to certain algebraic groups. On the basis of the method of IT-reduction a finite generation theorem is proved: for tensor bundles $\mathscr{V},\mathscr{W}\to M$ all the natural differential operations $D\colon\Gamma(\mathscr{V})\to\Gamma(\mathscr{W})$ of degree at most $d$ can be algebraically constructed from some finite set of such operations. Conceptual proofs of known results on the classification of natural linear operations on arbitrary and symplectic manifolds are presented. A non-existence theorem is proved for natural deformation quantizations on Poisson manifolds and symplectic manifolds.
Bibliography: 21 titles.

DOI: https://doi.org/10.4213/sm3935

Full text: PDF file (642 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2008, 199:10, 1481–1503

Bibliographic databases:

UDC: 514.74+512.815.7
MSC: Primary 58A32, 53D55; Secondary 15A72, 81S10
Received: 12.08.2007

Citation: P. I. Katsylo, D. A. Timashev, “Natural differential operations on manifolds: an algebraic approach”, Mat. Sb., 199:10 (2008), 63–86; Sb. Math., 199:10 (2008), 1481–1503

Citation in format AMSBIB
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    1. E. V. Ponomareva, “Classification of double flag varieties of complexity 0 and 1”, Izv. Math., 77:5 (2013), 998–1020  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. E. G. Puninskiy, “Natural operators on tensor fields”, Moscow University Mathematics Bulletin, 69:5 (2014), 225–228  mathnet  crossref  mathscinet
    3. D. A. Timashev, “On differential characteristic classes of metrics and connections”, J. Math. Sci., 223:6 (2017), 763–774  mathnet  crossref  mathscinet  elib
    4. Navarro A., Navarro J., Prieto C.T., “Natural Operations on Holomorphic Forms”, Arch. Math.-Brno, 54:4 (2018), 239–254  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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