A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the behavior of Gaussian curvature at a non-degenerate lightlike point of a mixed type surface. To characterize the boundedness of Gaussian curvature at a non-degenerate lightlike points, we introduce several fundamental invariants along non-degenerate lightlike points, such as the lightlike singular curvature and the lightlike normal curvature. Moreover, using the results by Pelletier and Steller, we obtain the Gauss-Bonnet type formula for mixed type surfaces with bounded Gaussian curvature.

中文翻译：

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三维洛伦兹流形中具有有界高斯曲率的混合型曲面

混合型曲面是洛伦兹三流形中的连接规则曲面，具有非空的类时空点集。混合型曲面的诱导度量是一个特征变化度量，它们的轻点可以看作是这些度量的奇异点。在本文中，我们研究了混合型曲面的非退化类光点处的高斯曲率行为。为了表征非退化类光点处高斯曲率的有界性，我们沿非退化类光点引入了几个基本不变量，例如类光奇异曲率和类光法曲率。此外，使用 Pelletier 和 Steller 的结果，我们获得了具有有界高斯曲率的混合型曲面的 Gauss-Bonnet 型公式。