Abstract
In this paper, we study a class of singular Riemannian manifolds. The singular set itself is a smooth manifold with a cone-like neighborhood. By imposing a reasonable convergence condition on the metric, we can determine the local geometrical structure near the singular set. In general, the curvature near the singular set is unbounded. We prove that a bounded curvature assumption would have a strong implication on the geometrical and topological structures near the singular set. We also establish the Gauss–Bonnet–Chern formula, which can be applied to the study of singular Eistein 4-manifolds.
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Liu, ZD., Shen, Z. Riemannian Geometry of Conical Singular Sets. Annals of Global Analysis and Geometry 16, 29–62 (1998). https://doi.org/10.1023/A:1006597812394
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DOI: https://doi.org/10.1023/A:1006597812394