Local flatness

In topology, a branch of mathematics, local flatness is a property of a submanifold in a topological manifold of larger dimension. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Local flatness and the topology of ridge networks is of importance in the study of crumpled structures with importance in materials processing and mechanical engineering.

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 a standard inclusion of ${\displaystyle \mathbb {R} ^{d}}$ as a subspace of ${\displaystyle \mathbb {R} ^{n}}$. That is, there exists a homeomorphism ${\displaystyle U\to \mathbb {R} ^{n}}$ such that the image of ${\displaystyle U\cap N}$ coincides with ${\displaystyle \mathbb {R} ^{d}}$.

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. In more detail, we can set ${\displaystyle \mathbb {R} _{+}^{n}=\{y\in \mathbb {R} ^{n}\colon y_{n}\geq 0\}}$ and ${\displaystyle \mathbb {R} ^{d}=\{y\in \mathbb {R} ^{n}\colon y_{d+1}=\cdots =y_{n}=0\}}$.

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.

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).