We consider the following setting: suppose that we are given a manifold M in $${\mathbb {R}}^d$$ with positive reach. Moreover assume that we have an embedded simplical complex $${\mathcal {A}}$$ without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in $${\mathcal {A}}$$ have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then $${\mathcal {A}}$$ is a triangulation of the manifold, that is, they are homeomorphic.

中文翻译：

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欧几里德空间子流形三角剖分的局部条件

我们考虑以下设置：假设我们在 $${\mathbb {R}}^d$$ 中给定了一个流形 M，其具有正范围。此外，假设我们有一个无边界的嵌入单纯复形 $${\mathcal {A}}$$，其顶点集位于流形上，并且足够密集，使得 $${\mathcal {A}}$ 中的所有单纯形$ 有足够的质量。我们证明，如果单纯形投影到切空间的局部内部不相交，则 $${\mathcal {A}}$$ 是流形的三角剖分，即它们是同胚的。