We study Dirac structures associated with Manin pairs (d, g) and give a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach [3] in terms of quasi-Poisson structures. We explain how these two distinct frameworks are related to each other, proving that they lead to isomorphic categories of Hamiltonian spaces. We stress the connection between the viewpoint of Dirac geometry and equivariant differential forms. The paper discusses various examples, including q-Hamiltonian spaces and Poisson-Lie group actions, explaining how presymplectic groupoids are related to the notion of “double” in each context.

In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing property defined and studied previously by J.E. D’Atri and H.K. Nickerson. In the last part of the paper we study properties of Clifford-Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres,and discuss numerous connections between these spaces and various classical objects.

For a compact Riemann surface X of positive genus, the space of sections of a certain theta bundle on moduli of bundles of rank r and level k admits a natural map to (the dual of) a similar space of sections of rank k and level r (the strange duality isomorphism). Both sides of the isomorphism carry projective connections as X varies in a family. We prove that this map is (projectively) flat.

In this paper we explicitly construct Moishezon twistor spaces on nCP2 for arbitrary n ≥ 2 which admit a holomorphic Caction. When n = 2, they coincide with Y. Poon’s twistor spaces. When n = 3, they coincide with the ones studied by the author in [14]. When n ≥ 4, they are new twistor spaces, to the best of the author’s knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP4. It turns out that the structure of the minitwistor space is independent of n. Further, we explicitly construct a CP2-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP1 ×CP1 in LeBrun’s case is replaced by our minitwistor spaces found in [15].

Let be the scattering relation on a compact Riemannian manifold M with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let . be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry, by knowledge of 冃 and . restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from D covers T M; and that those geodesics have no conjugate points. Under an additional topological assumption, we prove that 冃 and . restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.