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 ${\displaystyle U\to R^{n}}$ such that the image of ${\displaystyle U\cap N}$ coincides with ${\displaystyle \mathbb {R} ^{d}}$.

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