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<p>We show that the Lipschitz-Free Space over a connected orientable <i>n</i>-dimensional Riemannian manifold <i>M</i> is isometrically isomorphic to a quotient of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^1(M,TM)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>T</mi> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the integrable sections of the tangent bundle <i>TM</i>, if <i>M</i> is either complete or lies isometrically inside a complete manifold <i>N</i>. 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 <i>M</i> is simply connected, one may replace “exact” with “closed” currents.</p>

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

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  • 发表日期：

    2025年12月17日

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  • Franz Luggin

摘要

We show that the Lipschitz-Free Space over a connected orientable n-dimensional Riemannian manifold M is isometrically isomorphic to a quotient of \(L^1(M,TM)\) L 1 ( M , T M ) , 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.