Matematicheskii Sbornik
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

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

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

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
\Bibitem{KatTim08} \by P.~I.~Katsylo, D.~A.~Timashev \paper Natural differential operations on manifolds: an algebraic approach \jour Mat. Sb. \yr 2008 \vol 199 \issue 10 \pages 63--86 \mathnet{http://mi.mathnet.ru/msb3935} \crossref{https://doi.org/10.4213/sm3935} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2473812} \zmath{https://zbmath.org/?q=an:1160.58006} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2008SbMat.199.1481K} \elib{https://elibrary.ru/item.asp?id=20359288} \transl \jour Sb. Math. \yr 2008 \vol 199 \issue 10 \pages 1481--1503 \crossref{https://doi.org/10.1070/SM2008v199n10ABEH003969} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000262711500008} \elib{https://elibrary.ru/item.asp?id=13595859} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-66149094301} 

• http://mi.mathnet.ru/eng/msb3935
• https://doi.org/10.4213/sm3935
• http://mi.mathnet.ru/eng/msb/v199/i10/p63

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

Related presentations:

This publication is cited in the following articles:
1. E. V. Ponomareva, “Classification of double flag varieties of complexity 0 and 1”, Izv. Math., 77:5 (2013), 998–1020
2. E. G. Puninskiy, “Natural operators on tensor fields”, Moscow University Mathematics Bulletin, 69:5 (2014), 225–228
3. D. A. Timashev, “On differential characteristic classes of metrics and connections”, J. Math. Sci., 223:6 (2017), 763–774
4. Navarro A., Navarro J., Prieto C.T., “Natural Operations on Holomorphic Forms”, Arch. Math.-Brno, 54:4 (2018), 239–254
5. Gordillo-Merino A., Navarro J., Sancho P., “A Remark on the Invariant Theory of Real Lie Groups”, Colloq. Math., 156:2 (2019), 295–300
•  Number of views: This page: 542 Full text: 226 References: 44 First page: 15