There exist several known global implicit function theorems. Those results tend to be tailored for specific applications, and it is rather difficult to state a universally useful version.

One result that I find particularly helpful goes back to Hadamard (see, e.g., Chapt. 6 of "The Implicit Function Theorem" by Krantz and Parks):

Theorem. Let $M$ and $N$ be smooth, connected manifolds of dimension $d$ and let $f:M\to N$ be a $C^1$ function. If

• $f$ is proper (i.e. $f^{-1}(K)\subset M$ is compact whenever $K\subset N$ is compact),

• the Jacobian of $f$ vanishes nowhere on $M$, and
• $N$ is simply connected,

then $f$ is a homeomorphism.

You might be interested also in this paper by Rheinboldt. It contains some topological conditions on when the local solvability of the equation $$F(x,f(x,z))=z$$ leads to the global solvability.

There exist several known global implicit function theorems. Those results tend to be tailored for specific applications. It seems to be rather difficult to state a universally useful one-size-fits-all version.

One result that I find particularly helpful goes back to Hadamard (see, e.g., Chapt. 6 of "The Implicit Function Theorem" by Krantz and Parks):

Theorem. Let $M$ and $N$ be smooth, connected manifolds of dimension $d$ and let $f:M\to N$ be a $C^1$ mapping. If

• $f$ is proper (i.e. $f^{-1}(K)\subset M$ is compact whenever $K\subset N$ is compact),

• the Jacobian of $f$ vanishes nowhere on $M$, and
• $N$ is simply connected,

then $f$ is a homeomorphism.

You might be interested also in this paper by Rheinboldt. It contains some topological conditions on when the local solvability of the equation $$F(x,f(x,z))=z$$ leads to the global solvability.

There exist several known global implicit function theorems. Those results tend to be tailored for specific applications, and it is rather difficult to state a universally useful version.

One result that I find particularly helpful goes back to Hadamard (see, e.g., Chapt. 6 of "The Implicit Function Theorem" by Krantz and Parks):

Theorem. Let $M$ and $N$ be smooth, connected manifolds of dimension $d$ and let $f:M\to N$ be a $C^1$ function. If

• $f$ is proper (i.e. $f^{-1}(K)\subset M$ is compact whenever $K\subset N$ is compact),

• the Jacobian of $f$ vanishes nowhere on $M$, and
• $N$ is simply connected,

then $f$ is a homeomorphism.

There exist several known global implicit function theorems. Those results tend to be tailored for specific applications, and it is rather difficult to state a universally useful version.

One result that I find particularly helpful goes back to Hadamard (see, e.g., Chapt. 6 of "The Implicit Function Theorem" by Krantz and Parks):

Theorem. Let $M$ and $N$ be smooth, connected manifolds of dimension $d$ and let $f:M\to N$ be a $C^1$ function. If

• $f$ is proper (i.e. $f^{-1}(K)\subset M$ is compact whenever $K\subset N$ is compact),

• the Jacobian of $f$ vanishes nowhere on $M$, and
• $N$ is simply connected,

then $f$ is a homeomorphism.

You might be interested also in this paper by Rheinboldt. It contains some topological conditions on when the local solvability of the equation $$F(x,f(x,z))=z$$ leads to the global solvability.