Geodesic
Differential geometry of $(n-m)m$-dimensional complexes in $n$-dimensional projective space
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 1, Tome 220 (2023), pp. 17-27.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider an $(n-m)m$-dimensional complex in the projective space $P_n$. In the principal bundle associated with this complex, we construct a fundamental-group connection and calculate the curvature and torsion of this connection. We examine this complex by the Cartan–Laptev method. We prove that the fundamental object of the $1$th order of this complex is a pseudoquasitensor, the curvature is a pseudotensor, and the torsion is a geometric object only in combination with the connection subobject and the fundamental object. We perform the compositional framing of the $(n-m)m$-dimensional complex. Also, we prove that this framing induces connections of three types in the principal bundle associated with the complex.
Keywords: Cartan–Laptev method, complex, projective space, connection, curvature, compositional framing.
Mots-clés : torsion 
@article{INTO_2023_220_a1,
     author = {O. Belova},
     title = {Differential geometry of $(n-m)m$-dimensional complexes in $n$-dimensional projective space},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {17--27},
     publisher = {mathdoc},
     volume = {220},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2023_220_a1/}
}
TY  - JOUR
AU  - O. Belova
TI  - Differential geometry of $(n-m)m$-dimensional complexes in $n$-dimensional projective space
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2023
SP  - 17
EP  - 27
VL  - 220
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2023_220_a1/
LA  - ru
ID  - INTO_2023_220_a1
ER  - 
%0 Journal Article
%A O. Belova
%T Differential geometry of $(n-m)m$-dimensional complexes in $n$-dimensional projective space
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2023
%P 17-27
%V 220
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2023_220_a1/
%G ru
%F INTO_2023_220_a1
O. Belova. Differential geometry of $(n-m)m$-dimensional complexes in $n$-dimensional projective space. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 1, Tome 220 (2023), pp. 17-27. http://geodesic.mathdoc.fr/item/INTO_2023_220_a1/

[8] Belova O. O., “Svyaznost v rassloenii, assotsiirovannom s mnogoobraziem Grassmana”, Differ. geom. mnogoobr. figur., 2000, no. 31, 8–11

[9] Belova O. O., “Geometricheskaya kharakteristika indutsirovannykh svyaznostei tsentrirovannogo mnogoobraziya Grassmana”, Differ. geom. mnogoobr. figur., 2006, no. 37, 10–13

[10] Belova O. O., “Svyaznost 2-go tipa v rassloenii, assotsiirovannom s grassmanopodobnym mnogoobraziem tsentrirovannykh ploskostei”, Differ. geom. mnogoobr. figur., 2007, no. 38, 6–12

[11] Belova O. O., “Geometricheskaya kharakteristika indutsirovannykh svyaznostei grassmanopodobnogo mnogoobraziya tsentrirovannykh ploskostei”, Differ. geom. mnogoobr. figur., 2008, no. 39, 13–18

[12] Belova O. O., “Svyaznosti v rassloeniyakh, assotsiirovannykh s mnogoobraziem Grassmana i prostranstvom tsentrirovannykh ploskostei”, Fundam. prikl. mat., 14:2 (2008), 29\cyrdash67

[13] Belova O. O., “Grassmanopodobnoe mnogoobrazie tsentrirovannykh ploskostei”, Mat. zametki., 104:6 (2018), 812–822

[14] Belova O. O., “Reduktsiya rassloenii grassmanopodobnogo mnogoobraziya tsentrirovannykh ploskostei pri normalizatsii”, Itogi nauki i tekhn. Sovr. mat. prilozh. Temat. obz., 180 (2020), 3–8

[15] Bliznikas V. I., “Nekotorye voprosy geometrii giperkompleksov pryamykh”, Tr. geom. semin. VINITI., 6 (1974), 43–111

[16] Bubyakin I. V., “O stroenii pyatimernykh kompleksov dvumernykh ploskostei v proektivnom prostranstve $P^5$ s edinstvennym torsom”, Mat. zametki SVFU., 24:2 (2017), 3–12

[17] Bubyakin I. V., “O stroenii kompleksov $m$-mernykh ploskostei proektivnogo prostranstva $P^n$, soderzhaschikh konechnoe chislo torsov”, Mat. zametki SVFU., 24:4 (2017), 3–16

[18] Zhovtenko O. M. Rol osnascheniya Bortolotti kongruentsii ploskostei, Differ. geom. mnogoobr. figur., 2000, no. 31, 31–36

[19] Konnov V. V. Ob odnom uslovii redutsiruemosti glavnykh rassloenii i ego primenenii v proektivnoi geometrii podmnogoobrazii, Fundam. prikl. mat., 7:4 (2001), 1003–1035

[20] Kuleshov A. V., “Obobschennye svyaznosti na komplekse tsentrirovannykh ploskostei v proektivnom prostranstve”, Differ. geom. mnogoobr. figur., 2010, no. 41, 75–85

[21] Laptev G. F., “Differentsialnaya geometriya pogruzhennykh mnogoobrazii. Teoretiko-gruppovoi metod differentsialno-geometricheskikh issledovanii”, Tr. Mosk. mat. o-va., 2 (1953), 275–382

[22] Lumiste Yu. G., “Indutsirovannye svyaznosti v pogruzhennykh proektivnykh i affinnykh rassloeniyakh”, Uch. zap. Tartus. un-ta., 177 (1965), 6–41

[23] Nikitina E. S., Bubyakin I. V., “K geometrii mnogoobraziya Segre $S(m,n)$”, Mat. zametki SVFU., 2004, 57–62

[24] Polyakova K. V., “Svyaznosti v rassloeniyakh, assotsiirovannykh s mnogoobraziem par kasatelnoi i soprikasayuscheisya ploskostei poverkhnosti”, Tr. geom. sem. Kazan. un-ta., 23 (1997), 99–112

[25] Polyakova K. V., “Parallelnye pereneseniya na poverkhnosti proektivnogo prostranstva”, Fundam. prikl. mat., 14:2 (2008), 129–177

[26] Polyakova K. V., “Obobschenie derivatsionnykh formul proektivnogo prostranstva”, Differ. geom. mnogoobr. figur., 2009, no. 40, 109–117

[27] Polyakova K. V., “Tangentsialnoznachnye formy 2-go poryadka”, Mat. zametki., 105:1 (2019), 84–94

[28] Polyakova K. V., Shevchenko Yu. I., “Sposob Lapteva—Lumiste zadaniya svyaznosti i gorizontalnye vektory”, Differ. geom. mnogoobr. figur., 2012, no. 43, 114–121

[29] Safonov D. A., “Obobschennaya affinnaya svyaznost i ee vyrozhdenie v affinnuyu svyaznost”, Differ. geom. mnogoobr. figur., 2014, no. 45, 120–125

[30] Shevchenko Yu. I., “Ob osnascheniyakh mnogoobrazii ploskostei v proektivnom prostranstve”, Differ. geom. mnogoobr. figur., 1978, no. 9, 124–133

[31] Shevchenko Yu. I., Osnascheniya golonomnykh i negolonomnykh gladkikh mnogoobrazii, KGU, Kaliningrad, 1998

[32] Shevchenko Yu. I., “Osnascheniya podmnogoobrazii golonomnogo i negolonomnogo tsentroproektivnoykh mnogoobrazii”, Differ. geom. mnogoobr. figur., 1997, no. 28, 86–98

[33] Shevchenko Yu. I., Osnascheniya tsentroproektivnykh mnogoobrazii, KGU, Kaliningrad, 2000

[34] Shevchenko Yu. I., “Priemy Lapteva i Lumiste zadaniya svyaznosti v glavnom rassloenii”, Differ. geom. mnogoobr. figur., 2006, no. 37, 185–193

[35] Shevchenko Yu. I., “Vyrozhdenie ploskostnoi affinnoi svyaznosti Stolyarova”, Fundam. prikl. mat., 16:2 (2010), 155–161

[36] Akivis M. A., Goldberg V. V., Projective Differential Geometry of Submanifolds, North-Holland, 1993

[37] Belova O., “The third type bunch of connections induced by an analog of Norden's normalization for the Grassmann-like manifold of centered planes”, Miskolc Math. Notes., 14:2 (2013), 557–560

[38] Belova O., “Generalized affine connections associated with the space of centered planes”, Mathematics., 9:7 (2021), 782

[39] Mikeš J. et al., Differential Geometry of Special Mappings, Univ. Palackého, Olomouc, 2015

[40] Polyakova K. V., “Prolongations generated by horizontal vectors”, J. Geom., 110 (2019), 53