Difference between revisions of "Smoothly compatible charts"

Latest revision as of 12:27, 12 November 2015

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

• Motivation for smooth structures
• Chart
• Transition map
• Topological manifold
• Atlas
• Smooth atlas

References

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