Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n). If $x\in N,$ we say N is locally flat at x if there is a neighborhood $U\subset M$ of x such that the topological pair$(U,U\cap N)$ is homeomorphic to the pair $(\mathbb {R} ^{n},\mathbb {R} ^{d})$, with the standard inclusion of $\mathbb {R} ^{d}\to \mathbb {R} ^{n}.$ That is, there exists a homeomorphism $U\to \mathbb {R} ^{n))$ such that the image of $U\cap N$ coincides with $\mathbb {R} ^{d))$. In diagrammatic terms, the following square must commute:

We call Nlocally flat in M if N is locally flat at every point. Similarly, a map $\chi \colon N\to M$ is called locally flat, even if it is not an embedding, if every x in N has a neighborhood U whose image $\chi (U)$ is locally flat in M.

In manifolds with boundary

The above definition assumes that, if M has a boundary, x is not a boundary point of M. If x is a point on the boundary of M then the definition is modified as follows. We say that N is locally flat at a boundary point x of M if there is a neighborhood $U\subset M$ of x such that the topological pair $(U,U\cap N)$ is homeomorphic to the pair $(\mathbb {R} _{+}^{n},\mathbb {R} ^{d})$, where $\mathbb {R} _{+}^{n))$ is a standard half-space and $\mathbb {R} ^{d))$ is included as a standard subspace of its boundary.

Consequences

Local flatness of an embedding implies strong properties not shared by all embeddings. Brown (1962) proved that if d = n − 1, then N is collared; that is, it has a neighborhood which is homeomorphic to N × [0,1] with N itself corresponding to N × 1/2 (if N is in the interior of M) or N × 0 (if N is in the boundary of M).