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

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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 
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     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},
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     url = {http://geodesic.mathdoc.fr/item/INTO_2023_220_a1/}
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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/