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SUMMARY:Jorge Lauret (Universidad Nacional de Córdoba)
DTSTART:20200812T230000Z
DTEND:20200812T235900Z
DTSTAMP:20260423T052931Z
UID:VSGS/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/VSGS/10/">Pr
 escribing Ricci curvature on homogeneous manifolds</a>\nby Jorge Lauret (U
 niversidad Nacional de Córdoba) as part of Virtual seminar on geometry wi
 th symmetries\n\n\nAbstract\nGiven a symmetric 2-tensor $T$ on a manifold 
 $M$\, it is a classical problem in Riemannian geometry to ask about the ex
 istence (and uniqueness) of a metric $g$ on $M$ such that  $\\textrm{Ric}(
 g) = T$  (see e.g. [Besse\,Chap.5]).  Assuming that $M$ is a homogeneous m
 anifold\, we will consider in the talk the $G$-invariant version of the pr
 oblem\, where $G$ is a (unimodular\, not necessarily compact) Lie group ac
 ting transitively on $M$.  \n\nAfter an overview of results and questions\
 , we will give a formula for the differential $d\\textrm{Ric}$ of the func
 tion $\\textrm{Ric}$ at a $G$-invariant metric $g$\, which is precisely th
 e Lichnerowicz Laplacian acting on $G$-invariant symmetric 2-tensors.  The
  formula is in terms of the moment map for the variety of Lie algebras.  \
 n\nAs an application\, we will consider the concept of Ricci local inverti
 bility for a metric $g$\, i.e.\, when the kernel of $d\\textrm{Ric}$ at $g
 $ consists only of the subspace generated by $g$.  This is equivalent to t
 he existence of a $G$-invariant solution $g'$ to the Prescribed Ricci Prob
 lem  $\\textrm{Ric}(g') = cT$  (for some $c>0$)\, for any $G$-invariant $T
 $ sufficiently close to $\\textrm{Ric}(g)$.  Our main result is that any i
 rreducible naturally reductive metric on $M$ with respect to $G$ is Ricci 
 locally invertible.       \n\nThis is joint work in progress with Cynthia 
 Will.\n
LOCATION:https://researchseminars.org/talk/VSGS/10/
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