The Splitting Theorem in Non-Smooth Context

We prove that an infinitesimally Hilbertian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper D left-parenthesis 0 comma upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">CD(0,N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space containing a line splits as the product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an infinitesimally Hilbertian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper D left-parenthesis 0 comma upper N minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">CD(0,N-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space. By ‘infinitesimally Hilbertian’ we mean that the Sobolev space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript 1 comma 2 Baseline left-parenthesis upper X comma sans-serif d comma German m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">d</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">m</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">W^{1,2}(X,\mathsf {d},\mathfrak {m})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.