Abstract

Let M be a smooth manifold of dimension n with two Riemannian metrics g₁, g₂ which are related by a²g₁ < g₂ < b²g₁. Let R^q be the Euclidean space with two Euclidean metrics h₁, h₂ such that h₁ −h₂ has distinct eigenvalues. Further, suppose that c²h₁ −h₂ is nondegenerate for each c in (a,b), and r_±(a²h₁ −h₂) ≥ 2n, where r₊ and r₋ denote respectively the positive and the negative ranks of an indefinite metric. Under these conditions we show that there exists an almost everywhere differentiable (Lipschitz) map f : M→R^q satisfying (df_x)^*h_i = g_i for i = 1,2 for almost all x in M.

Keywords

Lipschitz map, isometric immersion, convex integration

Mathematical Subject Classification

Primary: 26A16, 58J52

Authors