If ∇ is a torsionless connection on the tangent bundle of a manifold M the Weyl curvature W is the part of the curvature in kernel of the Ricci contraction. We give a coordinate free proof of Weyl’s result that W vanishes if and only if (M,∇) is (locally) diffeomorphic to RP with ∇, when transported to RP, in the projective class of ∇LC , the Levi-Civita connection of the Fubini–Study metric on...

If ∇ is a torsionless connection on the tangent bundle of a manifold M the Weyl curvature W is the part of the curvature in kernel of the Ricci contraction. We give a coordinate free proof of Weyl’s result that W vanishes if and only if (M,∇) is (locally) diffeomorphic to RP with ∇, when transported to RP, in the projective class of ∇LC , the Levi-Civita connection of the Fubini–Study metric on...

Abstract. A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. It is known that gradient maps of exponentially concave functions are solutions of a MongeKantorovich optimal transport problem and allow for a better gradient approximation than those of ordinary concave functions. The approximation error, called L-diver...

A local classification of all locally conformal flat Riemannian 4-manifolds whose Ricci tensor satisfies the equation ∇ ( ρ− 1 6 sg ) = 1 18 ds⊙ g as well as a local classification of all locally conformal flat Riemannian 4-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The problem of local descripti...

Full proofs of results announced here are to be found in [lO]. We consider the problem of classifying germs of neighbourhoods of locally flat topological submanifolds. Let M be a fixed topological manifold. An r-neighbourhood of M is a pair (i, N) where N is a topological manifold of dimension n+r and i:M—*N a locally flat embedding satisfying i^ÇdN) —dM. Denote the set of germs of r-neighbourh...

In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold (Mn, [g]). We prove that, on a closed and locally conformally flat manifold with...

of the Dissertation Ricci-Flat Anti-Self-Dual Asymptotically Locally Euclidean 4-Manifolds by Evan Patrick Wright Doctor of Philosophy in Mathematics Stony Brook University 2013 A classification result for Ricci-flat anti-self-dual asymptotically locally Euclidean 4-manifolds is obtained: they are either hyperkähler (one of the gravitational instantons classified by Kronheimer), or a cyclic quo...

Let L be a two component link in S, an embedding of two disjoint circles which is topologically locally flat, that is, which extends to an embedding of two solid tori. The link has Alexander polynomial one, if, and only if, the first homology of the universal abelian cover of the complement of the link vanishes. If the link has Alexander polynomial one, then the linking number of the two compon...