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

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

In manifolds with boundary edit

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 ${\displaystyle U\subset M}$  of x such that the topological pair ${\displaystyle (U,U\cap N)}$  is homeomorphic to the pair ${\displaystyle (\mathbb {R} _{+}^{n},\mathbb {R} ^{d})}$ , where ${\displaystyle \mathbb {R} _{+}^{n}}$  is a standard half-space and ${\displaystyle \mathbb {R} ^{d}}$  is included as a standard subspace of its boundary.

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).

• Euclidean space
• Neat submanifold

• Brown, Morton (1962), Locally flat imbeddings [sic] of topological manifolds. Annals of Mathematics, Second series, Vol. 75 (1962), pp. 331–341.
• Mazur, Barry. On embeddings of spheres. Bulletin of the American Mathematical Society, Vol. 65 (1959), no. 2, pp. 59–65. http://projecteuclid.org/euclid.bams/1183523034.