We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive extrinsic curvature and submanifolds with integrable relative conullity. We show that no complete submanifold of hyperbolic space with sufficiently high index of relative nullity has extrinsic geometry bounded away from zero. As an application of these results, we derive an interesting corollary for complete submanifolds of hyperbolic space with nonpositive extrinsic curvature and discourse on their relation to Milnor’s conjecture about complete surfaces with second fundamental form bounded away from zero. Finally, we also prove that every complete Euclidean submanifold with integrable relative conullity is a cylinder over the relative conullity.

中文翻译：

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在空间形式中具有相对无效性的完全子流形

我们使用基于分裂张量的技术沿相对零点分布对 Codazzi 方程进行显式积分，并根据环境空间的雅可比张量表达第二种基本形式。这种方法使我们能够轻松地恢复文献中关于具有球体相对无效的完整子流形的几个重要结果，并在双曲和欧几里得空间中推导出新的强结果。我们的主要定理的结果包括具有足够高的相对无效指数的子流形、具有非正外在曲率的子流形和具有可积相对共性的子流形的结果。我们表明，没有具有足够高的相对无效指数的双曲空间的完整子流形具有远离零的外在几何形状。作为这些结果的应用，我们为具有非正外曲率的双曲空间的完全子流形导出了一个有趣的推论，并讨论了它们与 Milnor 关于具有远离零的第二基本形式的完整曲面的猜想的关系。最后，我们还证明了每个具有可积相对共性的完整欧几里德子流形都是相对共性上的圆柱体。