Handbook of normal frames and coordinates

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This book offers a comprehensive survey of the theory of normal frames and coordinates in differential geometry, presenting both established results and new findings by the author. It serves as a reference manual and an introduction to current research, appealing to graduate and post-graduate students, as well as scientists in differential geometry and theoretical/mathematical physics. The bibliography includes standard textbooks and journal articles, reflecting the diverse audience. This monograph is the first to compile known facts about normal frames and coordinates into a single publication, providing detailed considerations and proofs. Normal coordinates or frames are defined as those in which the coefficients of a linear connection vanish on a subset, typically a submanifold, of a differentiable manifold. Previously, the existence of normal frames was established only for symmetric linear connections on submanifolds. Recent developments have expanded the understanding of normal frames for derivations of the tensor algebra over differentiable manifolds, covering arbitrary linear connections. These rigorous results hold significance for various physical applications, enhancing the relevance of the theory in practical contexts.