Varifold

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In mathematics, a varifold is, loosely speaking, a measure-theoretic generalization of the concept of a differentiable manifold, by replacing differentiability requirements with those provided by rectifiable sets, while maintaining the general algebraic structure usually seen in differential geometry. More closely, varifolds generalize the ideas of a rectifiable current. Varifolds are the topic of study in geometric measure theory.

[edit] Definition

Given an open subset $Ω$ of Euclidean space $\scriptstyle\mathbb{R}^n$ , an m-dimensional varifold on $Ω$ is defined as a Radon measure on the set

$\Omega \times G(n,m)$

where $G (n, m)$ is the Grassmannian of all m-dimensional linear subspaces of an n-dimensional vector space. The Grassmannian is used to allow the construction of analogs to differential forms as duals to vector fields in the approximate tangent space of the set $Ω$ .