Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce the concepts of these submanifolds into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds in Hilbert spaces. We also mention a classification problem of minimal orbits in hyperpolar PF actions on Hilbert spaces.

中文翻译：

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Hilbert空间中PF子流形的严格和干旱性质

严格子流形和干旱子流形分别构成有限维黎曼流形中的两类不同的最小子流形。在本文中，我们将这些子流形的概念引入希尔伯特空间中的一类适当的Fredholm（PF）子流形，讨论它们之间的关系，并展示希尔伯特空间中无穷维Austere PF子流形和干旱PF子流形的实例。我们还提到了希尔伯特空间上超极性PF动作中最小轨道的分类问题。