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SUMMARY:Parallel spinors on Riemannian and Lorentzian manifolds
DTSTART;VALUE=DATE-TIME:20190322T130000Z
DTEND;VALUE=DATE-TIME:20190322T150000Z
DTSTAMP;VALUE=DATE-TIME:20220123T203109Z
UID:indico-event-4483@indico.math.cnrs.fr
DESCRIPTION:The talk describes results in joint articles with Klaus Krönc
ke\, Olaf Müller\, Hartmut Weiss\, and Frederik Witt.\n \nWe say that a
Riemannian metric on \\(M\\) is structured if its pullback to the universa
l cover admits a parallel spinor. All such metrics are Ricci-flat. The hol
onomy of these metrics is special as these manifolds carry some additional
structure\, e.g. a Calabi-Yau structure or a \\(G_2\\)-structure. All k
nown compact Ricci-flat manifolds are structured.\n\nThe set of structured
Ricci-flat metrics on compact manifolds is now well-understood\, and we w
ill explain this in the first part of the talk.\n\nThe set of structured R
icci-flat metrics is an open and closed subset in the space of all Ricci-f
lat metrics. The holonomy group is constant along connected components. Th
e dimension of the space of parallel spinors as well. The structured Ricci
-flat metrics form a smooth Banach submanifold in the space of all metrics
. Furthermore the associated premoduli space is a finite-dimensional smoot
h manifold\, and the parallel spinors form a natural bundle with metric an
d connection over this premoduli space.\n\nLorentzian manifolds with a par
allel spinor are not necessarily Ricci-flat\, however the rank of the Ricc
i tensor is at most \\(1\\)\, the image of the Ricci-endomorphism is light
like. Helga Baum\, Thomas Leistner and Andree Lischewski showed the well-p
osedness for an associated Cauchy problem. Here well-posedness means tha
t a (local) solutions exist if and only if the initial conditions satisfy
some constraint equations.\n\nWe are now able to prove a conjecture by Lei
stner and Lischewski which states that solutions of the constraint equatio
ns on an \\(n\\)-dimensional Cauchy hypersurface can be obtained from curv
es in the moduli space of structured Ricci-flat metrics on an \\((n-1)\\)-
dimensional closed manifold.\n\nhttps://indico.math.cnrs.fr/event/4483/
LOCATION:Tours 1180 (Bât E2)
URL:https://indico.math.cnrs.fr/event/4483/
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