On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds

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On the Conformal Geometry of Transverse
Riemann-Lorentz Manifolds

• V. Fernández
• Universidad Complutense de Madrid

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Transverse Riemann-Lorentz Manifolds
M connected manifold symmetric (0,2) tensor
field on M
? p?M degenerates?ø Radp(M)
?ø, p?? M-? is a union of
pseudoriemannian manifolds
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DEF M is a transverse type-changing manifold if
for every p??
? is a hypersurface that locally separates two
open pseudoriemannian manifolds whose indices
differ in one and dim(Radp(M))1
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DEF M is a transverse Riemann-Lorentz manifold
if the components of M-? are either riemannian or
lorentzian.
Example
? type-changing hypersurface
M riemannian
M- lorentzian
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On M-? we have all the geometric objects
associated to g derived from the Levi-Civita
connection D covariant derivatives, parallel
transport, geodesics, curvatures
PROBLEM Extendability to M of these objects?
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? the unique torsion-free and metric dual
connection on M such that on M-?
R a radical vectorfield (Rp?Radp(M)-0)
well-defined map
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DEF M is II-FLAT if
OBS M is II-flat iff extends to M
whenever or are tangent to ?.
DEF M is III-FLAT if
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THEOREM (Kossowski,97) K covariant curvature
extends to M iff radical transverse to ? and M
II-flat. Ric Ricci tensor extends to M iff
radical transverse to ? and M III-flat.
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Conformal Geometry

• transverse Riemann-Lorentz

C conformal structure
is also transverse Riemann-Lorentz whith
the same type-changing hypersurface ? and same
radical Rad
DEF conformal transverse
Riemann-Lorentz manifold
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Weyl Conformal Curvature

• DEF
• N pseudoriemannian manifold
• Weyl tensor of N, where
• Schouten tensor
• Kulkarni-Nomizu product

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OBS
thus
conformal invariant, called
Weyl conformal curvature
THEOREM (Weyl, 1918) iff M conformally
flat
that is, around every p?N there exists a metric
on the conformal structure which is flat
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Extendability of Weyl tensor

• THEOREM
• W extends to the whole M iff
• radical transverse and
• M conformally III-flat

that is, around every p?? there exists a metric
on the conformal structure which is III-flat