Definition:Compatible Charts

Definition

Let $M$ be a topological space.

Let $d$ be a natural number.

Let $\struct {U, \phi}$ and $\struct {V, \psi}$ be $d$-dimensional charts of $M$.

Then $\struct {U, \phi}$ and $\struct {V, \psi}$ are $C^k$-compatible if and only if their transition mapping:

$\psi \circ \phi^{-1}: \map \phi {U \cap V} \to \map \psi {U \cap V}$

is of class $C^k$.

Smoothly Compatible Charts

$\struct {U, \phi}$ and $\struct {V, \psi}$ are smoothly compatible if and only if their transition mapping:

$\psi \circ \phi^{-1} : \map \phi {U \cap V} \to \map \psi {U \cap V}$

is of class $C^\infty$.

Also see

• Definition:Atlas