We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kähler manifold gives rise to a c-projective structure and this is one of the primary motivations for its study. The existence of two or more Kähler metrics underlying a given c-projective structure has many ramifications, which we explore in depth. As a consequence of this analysis, we prove the Yano-Obata Conjecture for complete Kähler manifolds: if such a manifold admits a one parameter group of c-projective transformations that are not affine, then it is complex projective space, equipped with a multiple of the Fubini-Study metric. Proof. By assumption, we haveIf ∇ a q = 0, it follows easily that p and q are locally constant hence constant. Otherwise, contracting this expression with a nonzero tangent vector X a in the kernel of ∇ a q, we deduce that Λ b = qΛ b and ∇ a p = −ξ∇ a q for some function ξ., it follows from what we have already proven that ξ is constant. Otherwise, we deduce that p and q are constant.

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For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein-Gelfand-Gelfand complex associated with the trivial representation. However, at least in the cases we discuss, our constructions are relatively simple and avoid most of the machinery of parabolic geometry. Moreover, our method extends to contact and symplectic geometries (beyond the parabolic realm).2000 Mathematics Subject Classification. 53A40, 53D10, 58A12, 58A17, 58J10, 58J70.

We study the local geometry of irreducible parabolic geometries admitting strongly essential flows; these are flows by local automorphisms with higher-order fixed points. We prove several new rigidity results, and recover some old ones for projective and conformal structures, which show that in many cases the existence of a strongly essential flow implies local flatness of the geometry on an open set having the fixed point in its closure. For almost c-projective and almost quaternionic structures we can moreover show flatness of the geometry on a neighborhood of the fixed point.Date: July 30, 2018. π → M, ω) be a normal Cartan geometry of irreducible parabolic type (g, P ). We denote by inf(M ) the algebra of smooth vector fields η ∈ X(M ) along which the flow {ϕ t η }, where defined, is by automorphisms of (B π → M, ω); one need not assume η complete. The unique lift of η to B is denotedη.The P -equivariance of the Cartan connection ω implies that a different choice of b 0 ∈ π −1 (x 0 ) yields a conjugate value for the isotropy.Remark that η is strongly essential if and only if its isotropy at x 0 with respect to any b 0 belongs to the nilradical p + of p. Via the duality p + ∼ = (g/p) * of P -modules, the isotropy of Karin Melnick