144,516 research outputs found
A conformally invariant differential operator on Weyl tensor densities
We derive a tensorial formula for a fourth-order conformally invariant
differential operator on conformal 4-manifolds. This operator is applied to
algebraic Weyl tensor densities of a certain conformal weight, and takes its
values in algebraic Weyl tensor densities of another weight. For oriented
manifolds, this operator reverses duality: For example in the Riemannian case,
it takes self-dual to anti-self-dual tensors and vice versa. We also examine
the place that this operator occupies in known results on the classification of
conformally invariant operators, and we examine some related operators.Comment: 17 pages, LaTe
On the geometry of cohomogeneity one manifolds with positive curvature
This is a survey on cohomogeneity one manifolds with positive curvature. We
discuss the known examples of this type and their geometry and the functions
that describe the metric. We also describe the classification of cohomogeneity
one manifolds that can admit a metric with positive curvature due to
Grove-Wilking-Ziller. Two series of candidates arose in this classification for
which it is not yet know if they admit positive curvature. The connection of
these candidates to the Hitchin self dual Einstein orbifolds is discussed as
well, including the curvature properties of the Hitchin metrics.Comment: 25 page
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