Legendre Submanifolds

Abstract

In this chapter, we develop the framework necessary to study submanifolds within the context of Lie sphere geometry. The manifold Λ²ⁿ⁻¹ of projective lines on the Lie quadric Q ⁿ⁺¹ has a contact structure, i.e., a globally defined 1-form ω such that ω ∧ (dω)ⁿ⁻¹ ≠ 0 on Λ²ⁿ⁻¹. This gives rise to a codimension one distribution D on Λ²ⁿ⁻¹ which has integral submanifolds of dimension n−1, but none of higher dimension. These integral submanifolds are called Legendre submanifolds. Any submanifold of a real space-form ℝⁿ, S ⁿ or H ⁿ naturally induces a Legendre submanifold, and thus Lie sphere geometry can be used to analyze submanifolds of these spaces. This has been particularly effective in the classification of Dupin submanifolds, which are defined in Section 3.4. In Section 3.5, we define the Lie curvatures of a Legendre submanifold. These are natural Lie invariants which have proven to be valuable in the study of Dupin submanifolds but are defined on the larger class of Legendre submanifolds. We then give a Lie geometric characterization of those Legendre submanifolds which are Lie equivalent to an isoparametric hypersurface in a sphere (Theorem 5.6). In Section 3.6, we prove that tautness is Lie invariant. Finally, in Section 3.7, we discuss the counterexamples of Pinkall-Thorbergsson and Miyaoka-Ozawa to the conjecture that a compact proper Dupin hypersurface in a sphere must be Lie equivalent to an isoparametric hypersurface.