Difference between revisions of "Smoothly compatible charts"

Latest revision as of 12:27, 12 November 2015

Definition

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

• $U\cap V=\emptyset$
• $\psi\circ\varphi^{-1}$ is a Diffeomorphism

  That is:

  • $\psi\circ\varphi^{-1}$ is Smooth
  • $\psi\circ\varphi^{-1}$ is bijective
  • $\psi\circ\varphi^{-1}$'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. Jump up ↑ Introduction to smooth manifolds - John M Lee - Second Edition