Bismut Ricci flat generalized metrics on compact homogeneous spaces

2023/07/27 by Jorge Lauret, Cynthia Will · 1 citation
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Geometry and complex manifolds

paper · doi:10.1090/tran/9013

Abstract

A generalized metric on a manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , i.e., a pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis g comma upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(g,H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Riemannian metric and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a closed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -form, is a fixed point of the generalized Ricci flow if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis g comma upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(g,H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Bismut Ricci flat: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -harmonic and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R c left-parenthesis g right-parenthesis equals one fourth upper H Subscript g Superscript 2"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>g</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">Rc(g)=\tfrac 14H_g²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . On any homogeneous space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M equals upper G slash upper K"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">M=G/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals upper G 1 times upper G 2"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> × </mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">G=G₁× G₂</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant generalized metric, which is proved to be unique among a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -parameter space of metrics in many cases, including when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is neither abelian nor semisimple. On the other hand, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org

Cited by

• Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries

Related

• Rigidity of compact manifolds with boundary and nonnegative Ricci curvature
• Approximating topological metrics by Riemannian metrics
• Weighted monotonicity theorems and applications to minimal surfaces of ℍⁿ and 𝕊ⁿ
• Embedding 3-manifolds with circle actions
• A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below