arXiv subjects
Publications and source records attributed to Jack McKee.
Two generalizations of the Frobenius integrability theorem are proved concerning distributions which are piecewise-C1 but may fail to be continuous. The criteria presented are sufficient, but not necessary, for complete integrability of such distributions with bi-Lipschitz coordinates.
We use moving frame techniques to derive a notion of curvature for a class of piecewise-smooth Riemannian metrics called Regge metrics, showing that it is a measure that simultaneously satisfies the (weak) Cartan structure equations and the appropriate gauge transformation law. It turns out that this distributional curvature is equivalent to existing notions of densitized distributional curvature. We also investigate more closely the n = 2 case, where we prove the Gauss-Bonnet theorem for Regge metrics.
We present some aspects of the theory of finite element exterior calculus as applied to partial differential equations on manifolds, especially manifolds endowed with an approximate metric called a Regge metric. Our treatment is intrinsic, avoiding wherever possible the use of preferred coordinates or a preferred embedding into an ambient space, which presents some challenges but also conceptual and possibly computational advantages. As an application, we analyze and implement a method for computing an approximate Levi-Civita connection form for a disc whose metric is itself approximate.