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SUMMARY:Yusuke Sakane (Osaka Univ.)
DTSTART:20210929T130000Z
DTEND:20210929T140000Z
DTSTAMP:20260417T104858Z
UID:CHEM/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CHEM/12/">Ex
 istence and non-existence of Einstein metrics  on compact homogeneous  man
 ifolds</a>\nby Yusuke Sakane (Osaka Univ.) as part of Workshop on compact 
 homogeneous Einstein manifolds\n\n\nAbstract\nWe discuss existence and non
 -existence of Einstein metrics on  certain compact homogeneous manifolds $
 G/H$. \nStudy of existence and  non-existence of Einstein metrics  on $G/H
 $ is started by Wang and Ziller in 1986.   Park and speaker  studied some 
 spaces $G/H$ with three irreducible summands in 1997\,  and more general r
 esults for non-existence are obtained by  Böhm in 2005. In 2008\, Dickins
 on and Kerr studied existence and non-existence of Einstein metrics on $G/
 H$ with two irreducible summands. \n\nOne of the results of Böhm is that 
 compact homogeneous manifolds $G/H = SU(n+k_1+ \\cdots +k_p)/S(SO(n)U(1)\\
 times U(k_1) \\times\\cdots\\times U(k_p))$ (where $SO(n)U(1) \\subset U(n
 ) $) does not admit $G$-invariant Einstein metrics\,  if $n > (k_1+\\cdots
 +k_p)^2+2$. He has obtained the results by considering ``traceless" part o
 f Ricci tensor.  \n\nWe can apply his method for other compact homogeneous
  manifolds $G/H$.  \n\n1) For $G/H = Sp(n+k_1+ \\cdots +k_p)/SO(n)Sp(1)\\t
 imes Sp(k_1) \\times\\cdots\\times Sp(k_p)$ \n(where $SO(n)Sp(1) \\subset 
 Sp(n)$)\,  if $n \\geq  6 (k_1+\\cdots+k_p)$ ($k_i \\geq  1$)\, then \n$G/
 H$ does not admit $G$-invariant Einstein metrics\n\n2) For $G/H = SO(4 n+k
 _1+ \\cdots +k_p)/Sp(n)Sp(1)\\times SO(k_1) \\times\\cdots\\times SO(k_p)$
  \n(where $Sp(n)Sp(1) \\subset SO(4n)$)\, if  $n \\geq  (3/2)( k_1+ \\cdot
 s +k_p)$ ($k_i \\geq 3$)\, then $G/H$ does not admit $G$-invariant Einstei
 n metrics. \n\nFor \n$G/H = SO(2 n+k_1+ \\cdots +k_p)/ SO(n)U(1)\\times SO
 (k_1) \\times\\cdots\\times SO(k_p)$ (where $SO(n)U(1) \\subset U(n) \\sub
 set SO(2 n)$)\, we can not  apply the method directly. In this talk\, we w
 ill show that\, if $n \\geq 5 (k_1+ \\cdots +k_p)$ ($k_i \\geq 3$)\, $G/H$
  does not admit $G$-invariant Einstein metrics  by modifying argument of  
 ``traceless" part of Ricci tensor. \n\nFor existence of Einstein metrics\,
  in general\, it is difficult for compact homogeneous manifolds $G/H$ abov
 e.\n But\, in case of $ k_1 = \\cdots = k_p=k $\, we see that\, for a  giv
 en $p \\geq 1$ there exists a pair \n$(n\, k)$ such that  $G/H$ admits at 
 least two  $G$-invariant Einstein metrics.\n
LOCATION:https://researchseminars.org/talk/CHEM/12/
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