Developable hypersurfaces and homogeneous spaces in a real projective space. (English) Zbl 0955.53035

Ishikawa, G.

Lobachevskii J. Math. 3, 113-125 (1999).

Developable surfaces have been the object of study for some time, mainly for two reasons: they are surfaces generated by straight lines, and they seem to be ‘scarce’.
In fact, for dimension three in the real case, the only embedded developable surfaces are the cones, cylinders, and tangent developables of space curves [For an elementary treatment in \(\mathbb{R}^3\), see D. J. Struik, Lectures on Classical Differential Geometry, 2nd. ed. (1988; Zbl 0697.53002)]. There are also examples in \(\mathbb{R}^4\) [R. Sacksteder, Am. J. Math. 82, 609-630 (1960; Zbl 0194.22701); H. Wu, Int. J. Math. 6, 461-489 (1995; Zbl 0839.53004); J. J. Stoker, Differential Geometry (1989; Zbl 0718.53001)]. In other spaces, there are either existence theorems, or few examples.
The author accomplishes this. He gives new examples of non-singular developable hypersurfaces in \(\mathbb{R} P^n\). They turn out to be algebraic homogeneous spaces of groups \(SO(3), SU(3), Sp(3)\) and \(F_4\).
This very readable article has an extensive bibliography.

Reviewer: Lilia Del Riego (San Luis Potosí)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A20	Projective differential geometry
53C30	Differential geometry of homogeneous manifolds
53A05	Surfaces in Euclidean and related spaces

Keywords:

projective duality; Cayley’s octonians; Veronese embedding; Jordan algebra; Monge-Ampère foliation; Severi variety; isoparametric hypersurface

Citations:

Zbl 0697.53002; Zbl 0194.22701; Zbl 0839.53004; Zbl 0718.53001