In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be differentiable manifolds. A function ${\displaystyle f:X\to Y}$ is a local diffeomorphism if, for each point ${\displaystyle x\in X}$, there exists an open set ${\displaystyle U}$ containing ${\displaystyle x}$ such that the image ${\displaystyle f(U)}$ is open in ${\displaystyle Y}$ and $${\displaystyle f\vert _{U}:U\to f(U)}$$ is a diffeomorphism.

A local diffeomorphism is a special case of an immersion ${\displaystyle f:X\to Y}$. In this case, for each ${\displaystyle x\in X}$, there exists an open set ${\displaystyle U}$ containing ${\displaystyle x}$ such that the image ${\displaystyle f(U)}$ is an embedded submanifold, and ${\displaystyle f|_{U}:U\to f(U)}$ is a diffeomorphism. Here ${\displaystyle X}$ and ${\displaystyle f(U)}$ have the same dimension, which may be less than the dimension of ${\displaystyle Y}$.

A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.

The inverse function theorem implies that a smooth map ${\displaystyle f:X\to Y}$ is a local diffeomorphism if and only if the derivative ${\displaystyle Df_{x}:T_{x}X\to T_{f(x)}Y}$ is a linear isomorphism for all points ${\displaystyle x\in X}$. This implies that ${\displaystyle X}$ and ${\displaystyle Y}$ have the same dimension.

It follows that a map ${\displaystyle f:X\to Y}$ between two manifolds of equal dimension (${\displaystyle \operatorname {dim} X=\operatorname {dim} Y}$) is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding), or equivalently, if and only if it is a smooth submersion. This is because, for any ${\displaystyle x\in X}$, both ${\displaystyle T_{x}X}$ and ${\displaystyle T_{f(x)}Y}$ have the same dimension, thus ${\displaystyle Df_{x}}$ is a linear isomorphism if and only if it is injective, or equivalently, if and only if it is surjective.

Here is an alternative argument for the case of an immersion: every smooth immersion is a locally injective function, while invariance of domain guarantees that any continuous injective function between manifolds of equal dimensions is necessarily an open map.

All manifolds of the same dimension are "locally diffeomorphic," in the following sense: if ${\displaystyle X}$ and ${\displaystyle Y}$ have the same dimension, and ${\displaystyle x\in X}$ and ${\displaystyle y\in Y}$, then there exist open neighbourhoods ${\displaystyle U}$ of ${\displaystyle x}$ and ${\displaystyle V}$ of ${\displaystyle y}$ and a diffeomorphism ${\displaystyle f:U\to V}$. However, this map ${\displaystyle f}$ need not extend to a smooth map defined on all of ${\displaystyle X}$, let alone extend to a local diffeomorphism. Thus the existence of a local diffeomorphism ${\displaystyle f:X\to Y}$ is a stronger condition than "to be locally diffeomophic." Indeed, although locally defined diffeomorphisms preserve differentiable structure locally, one must be able to "patch up" these (local) diffeomorphisms to ensure that the domain is the entire smooth manifold.