Inverting covariant exterior derivative

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Authors

KYCIA Radoslaw Antoni ŠILHAN Josef

Year of publication 2025
Type Article in Periodical
Magazine / Source Analysis and Mathematical Physics
MU Faculty or unit

Faculty of Science

Citation
web https://doi.org/10.1007/s13324-025-01085-3
Doi https://doi.org/10.1007/s13324-025-01085-3
Keywords Covariant exterior derivative; Poincar & eacute; lemma; Antiexact forms; Homotopy operator; Fibered set
Description The algorithm for inverting covariant exterior derivative is provided. It works for a sufficiently small star-shaped region of a fibered set - a local subset of a vector bundle and associated vector bundle. The algorithm contains some constraints that can fail, giving no solution, which is the expected case for parallel transport equations. These constraints are straightforward to obtain in the proposed approach. The relation to operational calculus and operator theory is outlined. The upshot of this paper is to show, using the linear homotopy operator of the Poincare lemma, that we can solve the covariant constant and related equations in a geometric and algorithmic way. The considerations related to the regularity of the solutions are provided.
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