Varifold

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.

Definition

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

:$Omega\; imes\; G(n,m)$

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 $Omega$.

References

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