Smoothly compatible charts

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Definition

Two charts, [ilmath](U,\varphi)[/ilmath] and [ilmath](V,\psi)[/ilmath] are said to be smoothly compatible[1] if we have either:

  • [ilmath]U\cap V=\emptyset[/ilmath]
  • [ilmath]\psi\circ\varphi^{-1} [/ilmath] is a Diffeomorphism
    That is:
    • [ilmath]\psi\circ\varphi^{-1} [/ilmath] is Smooth
    • [ilmath]\psi\circ\varphi^{-1} [/ilmath] is bijective
    • [ilmath]\psi\circ\varphi^{-1} [/ilmath]'s inverse is also Smooth


This is vital to define smooth atlases

See also

References

  1. ↑ Introduction to smooth manifolds - John M Lee - Second Edition
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