Transgression Forms In Dimension 4

Annali di Matematica Pura ed Applicata, 2004

A class of minimal almost complex submanifolds of a Riemannian manifold M 4n with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternionic Kähler manifoldM 4n of non zero scalar curvature, in particular, whenM 4 is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kähler submanifolds M 2n of maximal possible dimension 2n. More precisely, we prove that any such Kähler submanifold M 2n ofM 4n is the projection of a holomorphic Legendrian submanifold L 2n ⊂ Z of the twistor space (Z, H) ofM 4n , considered as a complex contact manifold with the natural holomorphic contact structure H ⊂ T Z. Any Legendrian submanifold of the twistor space Z is defined by a generating holomorphic function. This is a natural generalization of Bryant's construction of superminimal surfaces in S 4 = H P 1 .

Journal of Geometry, 1998

We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kähler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kähler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kähler.

Journal of Geometry, 1998

We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kähler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kähler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kähler.

Mathematische Annalen, 2006

Using the twistor correspondence, this article gives a oneto-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing theČech coboundary of 0-cochains on an elliptic curve covered by two annuli.

Rocky Mountain Journal of Mathematics, 2009

The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics h t compatible with the almost complex structures J 1 and J 2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In this paper we compute the first Chern form of the almost Hermitian manifold (Z, h t , J n), n = 1, 2 and find the geometric conditions on M under which the curvature of its Chern connection D n is of type (1, 1). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to D n and show that the Nijenhuis tensor of J 2 is D 2parallel provided the base manifold M is Einstein and self-dual.

Journal of Geometry and Physics, 2014

Transactions of the American Mathematical Society, 1999

On an almost quaternionic manifold (M 4n , Q) we study the integrability of almost complex structures which are compatible with the almost quaternionic structure Q. If n ≥ 2, we prove that the existence of two compatible complex structures I 1 , I 2 = ±I 1 forces (M 4n , Q) to be quaternionic. If n = 1, that is (M 4 , Q) = (M 4 , [g], or) is an oriented conformal 4-manifold, we prove a maximum principle for the angle function I 1 , I 2 of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure J on the twistor space Z of an almost quaternionic manifold (M 4n , Q) and show that J is a complex structure if and only if Q is quaternionic. This is a natural generalization of the Penrose twistor constructions.

Differential Geometry and its Applications, 2013

Given a special Kähler manifold (M, ω, J , ∇) we construct a subbundle of the generalized tangent bundle of M endowed with a natural special Kähler structure. Precisely we consider E = T (M) ⊕ T * (M) and the subbundle L ω = graph(ω); we prove that L ω is invariant with respect to the calibrated complex structure J g = O −g −1 g O of E defined by the Riemannian metric g = −ω J on M and we define a special connection ∇ on E by using a natural contravariant connection on T * (M) defined by ω. We prove that (L ω , (,) |Lω , J g |Lω , ∇ |Lω) is special Kähler, where (,) is the canonical symplectic structure on E. Moreover, by using the identification of T (M) ⊕ T * (M) with T (T * (M)) defined by the symplectic connection ∇, we describe the corresponding special Kähler subbundle of T (T * (M)). Also we prove that the construction is invariant with respect to the class of connections {∇ θ } introduced in Alekseevsky et al. [1].

Osaka Journal of Mathematics, 2008

We develop the twistor theory of G-structures for which the (linear) Lie algebra of the structure group contains an involution, instead of a complex structure. The twistor space Z of such a G-structure is endowed with a field of involutions J ∈ ¼(End T Z ) and a J -invariant distribution H Z . We study the conditions for the integrability of J and for the (para-)holomorphicity of H Z . Then we apply this theory to para-quaternionic Kähler manifolds of non-zero scalar curvature, which admit two natural twistor spaces (Z¯, J , H Z ),¯= ±1, such that J 2 =¯Id. We prove that in both cases J is integrable (recovering results of Blair, Davidov and Muskarov) and that H Z defines a holomorphic (¯= −1) or para-holomorphic (¯= +1) contact structure. Furthermore, we determine all the solutions of the Einstein equation for the canonical one-parameter family of pseudo-Riemannian metrics on Z¯.