We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $f:M_c^n\to {\mathbb{Q}}_{\tilde{c}}^{n+p}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature c into a space form of curvature $\tilde{c}$, and free of weak-umbilic points if $c>\tilde{c}$. We show that the substantial codimension of f is $p=n-1$ if, as shown by Cartan and Moore, the first normal bundle possesses the lowest possible rank $n-1$. These submanifolds are of a class that has been extensively studied due to their many properties. For instance, they are holonomic and admit Bäcklund and Ribaucour transformations.

中文翻译：

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关于空间形式的等曲率子流形

我们证明与E. Cartan和JD Moore的著名结果相反。让$F：{\mathrm{中号}}_C^{ñ}\to {\mathbb{问}}_{\stackrel{〜}{C}}^{ñ+p}$是具有恒定截面曲率c的黎曼流形的等距浸入曲率的空间形式$\stackrel{〜}{C}$，并且在没有弱脐点的情况下 $C>\stackrel{〜}{C}$。我们证明f的实质余维为$p=ñ-\mathrm{1个}$ 如Cartan和Moore所示，如果第一个法线束具有最低的等级 $ñ-\mathrm{1个}$。这些子流形由于其许多特性而被广泛研究。例如，它们是完整的，并接受Bäcklund和Ribaucour变换。