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W4385300845 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 {1}{4}H_g^2</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=G_1times G_2</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/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 simple and the standard metric is Einstein on both <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G 1 slash pi 1 left-parenthesis upper K right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G_1/pi _1(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G 2 slash pi 2 left-parenthesis upper K right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G_2/pi _2(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we give a one-parameter family of 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 metrics on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G slash upper K> <mml:semantics> <mml:mrow> <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>G/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M equals upper G times upper G slash normal upper Delta upper K> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mi>G</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 mathvariant=normal>Δ</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>M=Gtimes G/Delta K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M Superscript 35 Baseline equals upper S upper O left-parenthesis 8 right-parenthesis times upper S upper O left-parenthesis 7 right-parenthesis slash upper G 2> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>35</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>8</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>7</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>M^{35}=SO(8)times SO(7)/G_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
W4385300845 created "2023-07-28" @default.
W4385300845 creator A5020454082 @default.
W4385300845 creator A5029464712 @default.
W4385300845 date "2023-08-07" @default.
W4385300845 modified "2023-10-07" @default.
W4385300845 title "Bismut Ricci flat generalized metrics on compact homogeneous spaces" @default.
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W4385300845 doi "https://doi.org/10.1090/tran/9013" @default.
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