We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a certain genericity condition, or if the ambient space is a product of two CROSSes, we improve this to a lower bound on the index alone, which is affine in the first Betti number. To prove these results, we introduce a generalization of isometric immersions in Euclidean space. Compact symmetric spaces admit (and in fact are characterized by) such a structure with skew-symmetric second fundamental form.

中文翻译：

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紧凑对称空间中的虚拟沉浸和最小超曲面

我们表明，在紧凑的对称空间中，封闭的，浸入的，最小的超曲面满足索引的下限加上零值，该值线性依赖于它们的第一个Betti数。此外，如果最小超曲面满足某个一般性条件，或者如果环境空间是两个CROSS的乘积，则可以将其提高到单独的索引下限，即在第一个Betti数中是仿射的。为了证明这些结果，我们引入了欧氏空间中等距浸入的一般化。紧凑的对称空间允许（实际上以其为特征）具有偏斜第二基本形式的这种结构。