On the Riemannian curvature of a twistor space

2016

Isotropic almost complex structures induce a class of Riemannian metrics on tangent bundle of a Riemannian manifold. In this paper the curvature tensors of these metrics will be calculated.

Kodai Mathematical Journal, 2003

B.-Y. Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. The Lagrangian version of this inequality was proved by the same author. In this article, we obtain a sharp estimate of the Ricci tensor of a slant submanifold M in a complex space formM Mð4cÞ, in terms of the main extrinsic invariant, namely the squared mean curvature. If, in particular, M is a Kaehlerian slant submanifold which satisfies the equality case identically, then it is minimal. 1. Preliminaries Let M be a real n-dimensional submanifold of a complex m-dimensional complex space formM Mð4cÞ of constant holomorphic sectional curvature 4c. We denote by ' and' ' the Levi-Civita connections of M andM Mð4cÞ, respectively. Let J be the complex structure onM Mð4cÞ. Also, we denote by h the second fundamental form and R the Riemann curvature tensor of M.

International Journal of Geometric Methods in Modern Physics, 2014

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on CP 3 is a non-Kähler Hermitian-Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein-Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z. 190 (1985) 39-43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom. 7 (2007) 207-214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

Archiv der Mathematik, 2000

The famous Nash embedding theorem published in 1956 was aiming for the opportunity to use extrinsic help in the study of (intrinsic) Riemannian geometry, if Riemannian manifolds could be regarded as Riemannian submanifolds. However, this hope had not been materialized yet according to [23]. The main reason for this was the lack of control of the extrinsic properties of the submanifolds by the known intrinsic invariants. In order to overcome such difficulties as well as to provide answers to an open question on minimal immersions, the first author introduced in the early 1990's new types of Riemannian invariants, his so-called δ-curvatures, different in nature from the "classical" Ricci and scalar curvatures. One purpose of this article is to present some old and recent results concerning δ-invariants for Lagrangian submanifolds of complex space forms. Another purpose is to point out that the proof of Theorem 4.1 of [17] is not correct and the Theorem has to be reformulated. More precisely, Theorem 4.1 of [17] shall be replaced by Theorems 8.1 and 8.3 of this article. Since the new formulation needs a new proof, we also provide the proofs of Theorems 8.1 and 8.3 in this article.

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.

Proceedings of the American Mathematical Society, 1990

The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics ht compatible with the almost-complex structures J, and J2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In the present note we describe the (real-analytic) manifolds M for which the Ricci tensor of (Z , ht) is ./"-Hermitian, n = 1 or 2. This is used to supply examples giving a negative answer to the Blair and Ianus question of whether a compact almost-Kähler manifold with Hermitian Ricci tensor is Kählerian.

Mediterranean Journal of Mathematics, 2008

In this paper we study a Riemanian metric on the tangent bundle T (M ) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to T (M ) a structure of locally conformal almost Kählerian manifold. This is the natural generalization of the well known almost Kählerian structure on T (M ). We found conditions under which T (M ) is almost Kählerian, locally conformal Kählerian or Kählerian or when T (M ) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from T (M ). Moreover, we found that this map preserves also the natural almost contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively.

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.

Differential Geometry and its Applications, 2005

We study the Einstein condition for a natural family of Riemannian metrics on the twistor space of partially complex structures of a fixed rank on the tangent spaces of a Riemannian manifold compatible with its metric.