<p>We prove that there is a unique Levi-Civita connection on the one-forms of the Dabrowski-Sitarz spectral triple for the Podleś sphere <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5348_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(S^{2}_{q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>S</mi> <mi>q</mi> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation>. We compute the full curvature tensor, as well as the Ricci and scalar curvature of the Podleś sphere using the framework of Mesland and Rennie (Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms, Adv. Math. <b>468,</b> 110207 (2025)). The scalar curvature is a constant, and as the parameter <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5348_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(q\rightarrow 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the scalar curvature converges to the classical value 2. We prove a generalised Weitzenböck formula for the spinor bundle, which differs from the classical Lichnerowicz formula for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5348_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(q\ne 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≠</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, yet recovers it for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5348_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(q\rightarrow 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Curvature and Weitzenböck Formula for the Podleś Quantum Sphere

  • Bram Mesland,
  • Adam Rennie

摘要

We prove that there is a unique Levi-Civita connection on the one-forms of the Dabrowski-Sitarz spectral triple for the Podleś sphere \(S^{2}_{q}\) S q 2 . We compute the full curvature tensor, as well as the Ricci and scalar curvature of the Podleś sphere using the framework of Mesland and Rennie (Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms, Adv. Math. 468, 110207 (2025)). The scalar curvature is a constant, and as the parameter \(q\rightarrow 1\) q 1 , the scalar curvature converges to the classical value 2. We prove a generalised Weitzenböck formula for the spinor bundle, which differs from the classical Lichnerowicz formula for \(q\ne 1\) q 1 , yet recovers it for \(q\rightarrow 1\) q 1 .