The Sbrana–Cartan Hypersurfaces

Abstract

By the classical Beez-Killing theorem, a hypersurface $f\colon M^n\to \mathbb {Q}_c^{n+1}$ is rigid if it has type number τ ≥ 3 at any point. Therefore, if $f\colon M^n\to \mathbb {Q}_c^{n+1}$ is an isometric immersion such that M ⁿ admits another isometric immersion $\tilde f\colon M^n\to \mathbb {Q}_c^{n+1}$ that is not congruent to f on any open subset of M ⁿ, then f must have type number τ ≤ 2 at any point. Notice that f has type number τ ≤ 1 at a point of M ⁿ if and only if all sectional curvatures of M ⁿ at that point are equal to c, as follows from the Gauss equation. Totally geodesic hypersurfaces have already been classified in Chap. 1, whereas hypersurfaces of constant type number τ = 1 can locally be explicitly parametrized by means of the Gauss parametrization; see Corollaries 7.20 and 7.23.