Lipschitz solutions to the isometry relation for pairs of riemannian metrics

Giuseppina D’Ambra and Mahuya Datta

Vol. 237 (2008), No. 2, 223–240

DOI: 10.2140/pjm.2008.237.223

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 ∈ (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 ∈ M.

Keywords

Lipschitz map, isometric immersion, convex integration

Mathematical Subject Classification 2000

Primary: 26A16, 58J52

Milestones

Received: 19 July 2007

Revised: 29 February 2008

Accepted: 1 May 2008

Published: 1 October 2008