Isometric representation of Lipschitz-free spaces over connected orientable Riemannian manifolds

Abstract We show that the Lipschitz-Free Space over a connected orientable n -dimensional Riemannian manifold M is isometrically isomorphic to a quotient of $$L^1(M,TM)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , the integrable sections of the tangent bundle TM , if M is either complete or lies isometrically inside a complete manifold N . Two functions are deemed equivalent in this quotient space if their difference has distributional divergence zero. This quotient is the pre-annihilator of the exact essentially bounded currents, and if M is simply connected, one may replace “exact” with “closed” currents.