Diffeomorphism - Local Description

Local Description

Model example: if U and V are two connected open subsets of Rn such that V is simply connected, a differentiable map f: U → V is a diffeomorphism if it is proper and if

• the differential Df_x: Rn → Rn is bijective at each point x in U.

Remarks

• It is essential for U to be simply connected for the function f to be globally invertible (under the sole condition that its derivative is a bijective map at each point).
  • For example, consider the map (which is the "realification" of the complex square function) where U = V = R2 \ {(0,0)}. Then the map f is surjective and its satisfies (thus Df_x is bijective at each point) yet f is not invertible, because it fails to be injective, e.g., f(1,0) = (1,0) = f(-1,0).
• Since the differential at a point (for a differentiable function) is a linear map it has a well defined inverse if, and only if, Df_x is a bijection. The matrix representation of Df_x is the n × n matrix of first order partial derivatives whose entry in the i-th row and j-th colomn is . We often use this so-called Jacobian matrix for explicit computations.
• Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine that f were going from dimension n to dimension k. If n < k then Df_x could never be surjective, and if n > k then Df_x could never be injective. So in both cases Df_x fails to be a bijection.
• If Df_x is a bijection at x then we say that f is a local diffeomorphism (since by continuity Df_y will also be bijective for all y sufficiently close to x).
• Given a smooth map from dimension n to dimension k, if Df (resp. Df_x) is surjective then we say that f is a submersion (resp. local submersion), and if Df (resp. Df_x) is injective we say that f is an immersion (resp. local immersion).
• A differentiable bijection is not necessarily a diffeomorphism, e.g. f(x) = x3 is not a diffeomorphism from R to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of a homeomorphism that is not a diffeomorphism.
• f being a diffeomorphism is a stronger condition than f being a homeomorphism (when f is a map between differentiable manifolds). For a diffeomorphism we need f and its inverse to be differentiable. For a homeomorphism we only require that f and its inverse be continuous. Thus every diffeomorphism is a homeomorphism, but the converse is false: not every homeomorphism is a diffeomorphism.

Now, f: M → N is called a diffeomorphism if in coordinates charts it satisfies the definition above. More precisely, pick any cover of M by compatible coordinate charts, and do the same for N. Let φ and ψ be charts on M and N respectively, with U being the image of φ and V the image of ψ. Then the conditions says that the map ψ f φ−1: U → V is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every couple of charts φ, ψ of two given atlases, but once checked, it will be true for any other compatible chart. Again we see that dimensions have to agree.