Three-webs $W(r,r,2)$

A. M. Shelekhov

Sbornik. Mathematics, Tome 211 (2020) no. 6, pp. 875-899

Résumé

Local differential-geometric properties of three-webs $W(r,r,2)$ formed on a $2r$-dimensional manifold by foliations of codimension $r,r$ and $2$, respectively, are considered. In particular, three-webs defined by complex analytic functions of $r$ complex arguments belong to this class of webs. The structure equations of a three-web $W(r,r,2)$ in an adapted co-frame (in particular, in a natural co-frame) are deduced; the canonical connection $\Gamma$ on the manifold of a web $W(r,r,2)$ is introduced; formulae are obtained to calculate (in a natural co-basis) the components of the first structure tensor of a three-web $W(r,r,2)$ in terms of the derivatives of the function of this web. Three special classes of three-webs $W(r,r,2)$ are considered in detail: regular and group three-webs and also three-webs $W(r,r,2)$ generated by holomorphic functions. Bibliography: 17 titles.

Détail

Keywords: three-web $W(r,r,2)$, group three-web $W(r,r,2)$, regular three-web $W(r,r,2)$, three-web $\mathrm{CW}(r,r,2)$, canonical connection on a three-web $W(r,r,2)$.

Bibliographie

[1] Shiing-Shen Chern, “Abzählungen für Gewebe”, Abh. Math. Sem. Univ. Hamburg, 11:1 (1935), 163–170 | DOI | MR | Zbl

[2] M. A. Akivis, “O tri-tkanyakh mnogomernykh poverkhnostei”, Tr. Geom. semin., 2, VINITI, M., 1969, 7–31 ; РЖМат, 1970, 4А647 | MR | Zbl

[3] M. A. Akivis, A. M. Shelekhov, Geometry and algebra of multidimensional three-webs, Math. Appl. (Soviet Ser.), 82, Kluwer Acad. Publ., Dordrecht, 1992, xvii+358 pp. | DOI | MR | Zbl

[4] M. A. Akivis, A. M. Shelekhov, Mnogomernye tri-tkani i ikh prilozheniya, Tver. gos. un-t, Tver, 2010, 307 pp.

[5] M. A. Akivis, “Differentsialnaya geometriya tkanei”, Itogi nauki i tekhn. Ser. Probl. geom., 15, VINITI, M., 1983, 187–213 ; M. A. Akivis, “Differential geometry of webs”, J. Soviet Math., 29:5 (1985), 1631–1647 ; РЖМат, 1984, 7А630 | MR | Zbl | DOI

[6] M. A. Akivis, V. V. Goldberg, “Algebraic aspects of web geometry”, Comment. Math. Univ. Carolin., 41:2 (2000), 205–236 | MR | Zbl

[7] M. A. Akivis, V. V. Goldberg, “Differential geometry of webs”, Handbook of differential geometry, v. I, North-Holland, Amsterdam, 2000, 1–152 | DOI | MR | Zbl

[8] M. A. Akivis, V. V. Gol'dberg, “On multidimensional tri-nets formed by surfaces of different dimensions”, Soviet Math. Dokl., 13 (1972), 354–358 | MR | Zbl

[9] M. A. Akivis, V. V. Goldberg, “O mnogomernykh tri-tkanyakh, obrazovannykh poverkhnostyami raznykh razmernostei”, Tr. Geom. semin., 4, VINITI, M., 1973, 179–204 ; РЖМат, 1974, 5А372 | MR | Zbl

[10] V. V. Goldberg, “Transversalno-geodezicheskie, shestiugolnye i gruppovye tri-tkani, obrazovannye poverkhnostyami raznykh razmernostei”, Sbornik statei po differentsialnoi geometrii, Kalinin. gos. un-t, Kalinin, 1974, 52–69 ; РЖМат, 1975, 8А674 | MR

[11] N. Kh. Azizova (Selivanova), “O tkanyakh iz krivykh i poverkhnostei”, Uch. zap. MGPI, 374:1, Voprosy differentsialnoi geometrii (1970), 7–17 ; РЖМат, 1971, 8А509 | MR

[12] N. Kh. Selivanova (Azizova), “Intransitive families of transformations”, Soviet Math. (Iz. VUZ), 28:12 (1984), 87–91 ; РЖМат, 1985, 5А629 | MR | Zbl

[13] A. A. Duyunova, “O tri-tkanyakh $W(1,n,1)$ s nulevym pervym strukturnym tenzorom”, Izv. PGPU im. V. G. Belinskogo. Fiz.-matem. i tekh. nauki, 2011, no. 26, 82–88

[14] A. A. Duyunova, “O privedenii sistemy ODU k kanonicheskomu vidu”, Izv. PGPU im. V. G. Belinskogo. Fiz.-matem. i tekh. nauki, 2011, no. 26, 76–81

[15] A. A. Duyunova, “Three-webs $W(1,n,1)$ and associated systems of ordinary differential equations”, Russian Math. (Iz. VUZ), 56:2 (2012), 37–49 | DOI | MR | Zbl

[16] A. A. Duyunova, “Three-webs defined by a system of ordinary differential equations”, J. Math. Sci. (N.Y.), 177:5 (2011), 654–667 | DOI | MR | Zbl

[17] W. Blaschke, Einführung in die Geometrie der Waben, Birkhäuser Verlag, Basel, Stuttgart, 1955, 108 pp. | MR | MR | Zbl