Abstract
We consider an N-dimensional Riemannian manifold M and let gi be a basis at the point Pi(u1,…,uN) and gj be another basis at the other point Pj(u1,…,uN). Note that each such basis may only exist in a local neighborhood of the respective points, and not necessarily for the whole space. For each such point we may construct an embedded affine tangential manifold. The N-tuple of coordinates are invariant in any chosen basis; however, its components on the coordinates change as the coordinate system varies. Therefore, the relating components have to be taken into account by the coordinate transformations.
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Nguyen-Schäfer, H., Schmidt, JP. (2017). Elementary Differential Geometry. In: Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers. Mathematical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48497-5_3
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DOI: https://doi.org/10.1007/978-3-662-48497-5_3
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