582 research outputs found
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 -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 -manifolds from 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
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