The Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

Abstract

Let M_t be an isoparametric foliation on the unit sphere (Sⁿ⁻¹(1), g^st) with d principal curvatures. Using the spherical coordinates induced by M_t, we construct a Minkowski norm with the representation $F = r\sqrt {2f(t)}$, which generalizes the notions of (α, β)-norm and (α₁, α₂)-norm. Using the technique of the spherical local frame, we give an exact and explicit answer to the question when $F = r\sqrt {2f(t)}$ really defines a Minkowski norm. Using the similar technique, we study the Hessian isometry Φ between two Minkowski norms induced by M_t, which preserves the orientation and fixes the spherical ξ-coordinates. There are two ways to describe this Φ, either by a system of ODEs, or by its restriction to any normal plane for M_t, which is then reduced to a Hessian isometry between Minkowski norms on ℝ² satisfying certain symmetry and (d)-properties. When d > 2, we prove that this Φ can be obtained by gluing positive scalar multiplications and compositions of the Legendre transformation and positive scalar multiplications, so it must satisfy the (d)-property for any orthogonal decomposition ℝⁿ = V′ + V″, i.e., for any nonzero x = x′ + x″ and $\Phi (x) = \bar x = \bar x\prime + \bar x\prime \prime$ with $x\prime ,\bar x\prime \in {\bf{V}}\prime$ and x″, $x\prime \prime ,\bar x\prime \prime \in {\bf{V}}\prime \prime$, we have $g_x^{{F_1}}(x\prime \prime ,x) = g_{\bar x}^{{F_2}}(\bar x\prime \prime ,\bar x)$. As byproducts, we prove the following results. On the indicatrix (S_F, g), where F is a Minkowski norm induced by M_t and g is the Hessian metric, the foliation N_t = S_F ∩ ℝ_>0M₀ is isoparametric. Laugwitz Conjecture is valid for a Minkowski norm F induced by M_t, i.e., if its Hessian metric g is flat on ℝⁿ{0} with n > 2, then F is Euclidean.