2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/96465The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The existence of bundle coordinates normal at a given point and/or along injective horizontal path is proved. A necessary and sufficient condition of existence of bundle coordinates normal along injective horizontal mappings is derived.24 LaTeX pages. The packages AMS-LaTeX and amsfonts are required. In version 2 some results are generalized and proved under weaker conditions. For other papers on the same topic view the "publication" pages at http://theo.inrne.bas.bg/~bozho/Differential GeometryGeneral Relativity and Quantum CosmologyMathematical Physics57R25 (Primary) 53B05, 53C05, 55R99, 58A30 (Secondary)text