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Geometry of Fibered Graphs of Mappings

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Abstract

In this paper, we examine the differential-geometric aspect of constant-rank mappings of smooth manifolds based on the concept of a graph as a smooth submanifold in the space of the direct product of the original manifolds. The nonmaximality of the rank provides the fibered nature of the graph. A Riemannian structure on manifolds enriches the geometry of the graph, which now essentially depends on the induced field of the metric tensor; we characterize relatively affine, projective, and g-umbilical mappings. The final part of the paper is devoted to mappings of Euclidean spaces of the types described earlier in terms of V. T. Bazylev’s constructive graph.

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  1. Financial University under the Government of the Russian Federation, Moscow, Russia

    A. A. Rylov

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  1. A. A. Rylov
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Correspondence to A. A. Rylov.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 179, Proceedings of the International Conference “Classical and Modern Geometry” Dedicated to the 100th Anniversary of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 1, 2020.

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Rylov, A.A. Geometry of Fibered Graphs of Mappings. J Math Sci 276, 400–409 (2023). https://doi.org/10.1007/s10958-023-06756-9

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