Tubular hypersurfaces satisfying a basic equality.

*(English)*Zbl 0827.53045Let \(M^n\) be an \(n\)-dimensional submanifold of a real space form \(\widetilde {M}(c)\) with constant sectional curvature \(c\). In a previous paper, the author proved that the Riemannian invariant \(\delta_M\) of \(M^n\), defined by \(\delta_M (p) = \tau (p) - \inf K(p)\), where \(\inf K(p)\) is the smallest sectional curvature at \(p \in M^n\) and \(\tau = \sum_{i < j} K(e_i \wedge e_j)\) is the scalar curvature, satisfies the following inequality:
\[
\delta M \leq {n^2(n - 2) \over 2(n - 1)} |H |^2 + {1\over 2} (n + 1) (n -2) c,
\]
where \(H\) is the mean curvature vector. This inequality is sharp, and many nice classes of submanifolds realize equality in this inequality.

In the present paper, the author classifies tubular hypersurfaces of real space forms which realize the equality. For this purpose, first a description is given of the shape operators of tubular hypersurfaces. Next the classification of tubular hypersurfaces satisfying the equality is given, for Euclidean spaces, hyperbolic spaces and Euclidean spheres separately. In the final paragraph, the classification of isoparametric hypersurfaces which realize the equality is given. It turns out that all of them are special types of tubular hypersurfaces. In this way many nice classes of submanifolds are characterized again by a basic equality.

In the present paper, the author classifies tubular hypersurfaces of real space forms which realize the equality. For this purpose, first a description is given of the shape operators of tubular hypersurfaces. Next the classification of tubular hypersurfaces satisfying the equality is given, for Euclidean spaces, hyperbolic spaces and Euclidean spheres separately. In the final paragraph, the classification of isoparametric hypersurfaces which realize the equality is given. It turns out that all of them are special types of tubular hypersurfaces. In this way many nice classes of submanifolds are characterized again by a basic equality.

Reviewer: F.Dillen (Leuven)