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Weakly Differentiable Mappings between Manifolds

机译：流形之间的弱微分映射

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We study Sobolev classes of weakly differentiable mappings f : X — V between compact Riemannian manifolds without boundary. These mappings need not be continuous. They actually possess less regularity than the mappings in W~(1,n) (Х, Y) , n = dim X. The central themes being discussed are:smooth approximation of those mappingsintegrability of the Jacobian determinant The approximation problem in the category of Sobolev spaces between manifolds W~(1 P)(X, Y), 1 p z n, has been recently settled in [2], [3], [17], [23], [24]. However, the point of our results is that we make no topological restrictions on manifolds X and Y. We characterize, essentially all, classes of weakly differentiable mappings which satisfy the approximation property. The novelty of our approach is that we were able to detect tiny sets on which the mappings are continuous. These sets give rise to the so-called web like structure of % associated with the given mapping f : X —> Y.
机译：我们研究了无边界的紧凑黎曼流形之间的弱可微映射f：X-V的Sobolev类。这些映射不必是连续的。它们实际上比W〜（1，n）（Х，Y），n = dim X中的映射具有更少的规则性。所讨论的中心主题是：那些映射的平滑近似Jacobian行列式的可积性流形W〜（1 P）（X，Y），1 pzn之间的Sobolev空间最近在[2]，[3]，[17]，[23]，[24]中得到了解决。但是，我们的结果的重点是，我们对流形X和Y没有任何拓扑限制。我们基本上描述了满足逼近性质的所有类型的弱微分映射。我们方法的新颖之处在于我们能够检测到连续映射的微小集合。这些集合产生了与给定映射f：X-> Y相关的％的所谓的类似Web的结构。

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