About the geometry of almost para-quaternionic manifolds

We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the canonical almost complex structure of the twistor space and to the integrability of the canonical almost para-complex structure of the reflector space of (M,P). We show that (M, P) has plenty of locally defined, compatible, complex and para-complex structures, provided that P is para-quaternionic.Comment: 26 page

The Bochner-flat cone of a CR manifold

We construct a Kahler structure (which we call a generalised Kahler cone) on an open subset of the cone of a strongly pseudo-convex CR manifold endowed with a 1-parameter family of compatible Sasaki structures. We determine those generalised Kahler cones which are Bochner-flat and we study their local geometry. We prove that any Bochner-flat Kahler manifold of complex dimension bigger than two is locally isomorphic to a generalised Kahler cone.Comment: 29 page

Hermitian metrics on F-manifolds

An $F$-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated hypersurface singularities. This paper reviews the construction of hermitian metrics on $F$-manifolds from $tt^*$ geometry. It clarifies the logic between several notions. It also introduces a new {\it canonical} hermitian metric. Near irreducible points it makes the manifold almost hyperbolic. This holds for the singularity case and will hopefully lead to applications there.Comment: 2nd version 36 pages. Compared to the 1st version (32 pages), the sections 2.4 and 2.5 have been extende

Regular F-manifolds: initial conditions and Frobenius metrics

A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E), such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow T_{p}M. We obtain that any regular F-manifold admits a preferred system of local coordinates and we find conditions, in these coordinates, for a metric to be Frobenius. We study the Lie algebra of infinitesimal symmetries of regular F-manifolds. We show that any regular F-manifold is locally isomorphic to the parameter space of a Malgrange universal connection. We prove an initial condition theorem for Frobenius metrics on regular F-manifolds.Comment: 35 pages; with respect to the previous version, Section 4 is reorganised; reference [17] is added; other minor correction