1995/06/01 by Steven C. Ferry, Boris L. Okun, Boris Okun · 2 citations
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.1090/s0002-9939-1995-1246524-7
We study the relation between (topological) inner metrics and Riemannian metrics on smoothable manifolds. We show that inner metrics on smoothable manifolds can be approximated by Riemannian metrics. More generally, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper M right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f:M → X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a continuous surjection from a smooth manifold to a compact metric space with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript negative 1 Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f - 1(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> connected for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper X"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x ∈ X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then there is a metric <italic>d</italic> on <italic>X</italic> and a sequence of Riemannian metrics <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace psi Subscript i Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false"></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\ ψ i\</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <italic>M</italic> so that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M comma psi Subscript i Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(M,ψ i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> converges to ( <italic>X, d</italic> ) in Gromov-Hausdorff space. This is used to obtain a (fixed) contractibility function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi> ρ </mml:mi> <mml:annotation encoding="application/x-tex">ρ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a sequence of Riemannian manifolds with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi> ρ </mml:mi> <mml:annotation encoding="application/x-tex">ρ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as contractibility function so that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit left-parenthesis upper M comma psi Subscript i Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">lim (M,ψ i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is infinite dimensional. Using results of Dranishnikov and Ferry, this also gives examples of nonhomeomorphic manifolds <italic>M</italic> and <italic>N</italic> and a contractibility function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi> ρ </mml:mi> <mml:annotation encoding="application/x-tex">ρ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> ε </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">ε > 0</mml:annotation> </mml:semantics>