Mohammed L. Labbi, University of Bahrein
On compact manifolds with positive modified Einstein tensors
Abstract: Let M be a compact n-manifold and g be a Riemannian
metric on M. We introduce a metric invariant Ein(g)∊[0, n] that
measures in some sense the order of positivity of the scalar curvature
of g. It is positive if and only if g is a positive scalar curvature
metric and it is maximal equal to n if g is an Einstein metric with
positive scalar curvature. We introduce as well a conformal invariant
Ein([g]) and a smooth invariant Ein(M) to be respectively the
supremum of all the Ein(h) where h runs over all Riemannian metrics
in the conformal class of g and the supremum over all Riemannian
metrics on M.
In this talk, I will present several examples where one can compute
these invariants and then I will prove some general geometric
properties of these invariants and their interactions with the
topology of the manifold M.