For any real-analytic Ka Èhler manifold we shall prove the existence of a hyperka Èhler metric in a neighbourhood of the zero section of its cotangent bundle using the twistor method. This hyperka Èhler metric is compatible with the canonical holomorphicsymplectic structure of the cotangent bundle, extends the given Ka Èhler metric, and the circle action given by scalar multiplication in the ®bres is isometric.By deriving necessary conditions we show that many of these hyperka Èhler metrics will be incomplete.Furthermore, the S 1 -action given by scalar multiplication in the ®bres is isometric and the restriction of the hyperka Èhler metric to the zero section induces the original Ka Èhler metric.This result has been proved independently by D. Kaledin using di¨erent methods, cf.[13].Remark 1. This result also holds for real-analytic pseudo-Ka Èhler manifolds of sig-Brought to you by |
Using twistor techniques we shall show that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle T X of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1). The zero section is totally geodesic and the Obata connection restricts to the given connection on the zero section.We also prove an analogous result for vector bundles: any vector bundle with realanalytic connection whose curvature is of type (1, 1) over X can be extended to a hyperholomorphic bundle over a neighbourhood of the zero section of T X.
Abstract. Using the classification by Dotti and Fino [3] we show the existence of an HKT metric on a neighbourhood of the centre of any 8-dimensional nilpotent Lie group G with invariant hypercomplex structure. This metric exists globally if the hypercomplex structure is abelian, and in these cases we construct an HKT structure on a neighbourhood of the zero section of the cotangent bundle T * G extending the HKT metric on G.2000 Mathematics Subject Classification. 22E25.
Introduction.The two-dimensional sigma models studied by physicists force the Riemannian structure of the target space to be compatible with different kinds of quaternionic structures. In the presence of Wess Zumino terms and certain supersymmetries, the target space carries an HKT structure (see, for example, [9]). We begin by recalling some of the facts about these geometries.A manifold M is hypercomplex if there exists three complex structures I, J and K satisfying the relations of the quaternions I 2 = J 2 = K 2 = IJK = −1. A Riemannian manifold (M, g) is called hyperhermitian if it admits a hypercomplex structure such that g is hermitian with respect to I, J and K.An affine connection ∇ on a hyperhermitian manifold M is called hyperkähler with torsion if it satisfies ∇g = 0, ∇I = ∇J = ∇K = 0 and the torsion tensor c(X, Y, Z) = g(T(X, Y ), Z) is totally skew. (Here T is the torsion of ∇.) A manifold is called hyperkähler with torsion (or short HKT) if it is hyperhermitian and possesses a hyperkähler with torsion connection. It is well known (see [7]) that any hypercomplex manifold locally admits a compatible HKT metric.If there exists a hyperkähler with torsion connection on a hyperhermitian manifold, it is unique; see [7]. Hyperkähler manifolds with torsion are in general not hyperkähler as the Kähler forms corresponding to I, J and K need not be closed. The hyperkähler case corresponds to the case of vanishing torsion and the connection is then the LeviCivita connection.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.