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Register a new userNearly Kahler homogeneous manifolds with positive curvature
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We prove that a 2n-dimensional compact homogeneous nearly Kahler manifold with strictly positive sectional curvature is isometric to CP^{n}, equipped with the symmetric Fubini-Study metric or with the standard Sp(m)-homogeneous metric, n =2m-1, or to S^{6} as Riemannian manifold with constant sectional curvature. This is a positive answer for a revised version of a conjecture given by Gray.
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In this note we prove that a four-dimensional compact oriented half-confor-mally flat Riemannian manifold $M^4$ is topologically $mathbb{S}^{4}$ or $mathbb{C}mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval $[frac{3sqrt{3}-5}{4},,1].$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the $4$-sphere.
In this paper we study the class of compact Kahler manifolds with positive orthogonal Ricci curvature: $Ric^perp>0$. First we illustrate examples of Kahler manifolds with $Ric^perp>0$ on Kahler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of Kahler manifolds which admit metrics of $Ric^perp>0$. Secondly we prove some (algebraic) geometric consequences of the condition $Ric^perp>0$ to illustrate that the condition is also quite restrictive. Finally this last point is made evident with a classification result in dimension three and a partial classification in dimension four.
The moduli space NK of infinitesimal deformations of a nearly Kahler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly Kahler manifolds. It turns out that the nearly Kahler structure is rigid except for the flag manifold F(1,2)=SU_3/T^2, which carries an 8-dimensional moduli space of infinitesimal nearly Kahler deformations, modeled on the Lie algebra su_3 of the isometry group.
In this article we continue the study of the two curvature notions for Kahler manifolds introduced by the first named author earlier: the so-called cross quadratic bisectional curvature (CQB) and its dual ($^d$CQB). We first show that compact Kahler manifolds with CQB$_1>0$ or $mbox{}^d$CQB$_1>0$ are Fano, while nonnegative CQB$_1$ or $mbox{}^d$CQB$_1$ leads to a Fano manifold as well, provided that the universal cover does not contain a flat de Rham factor. For the latter statement we employ the Kahler-Ricci flow to deform the metric. We conjecture that all Kahler C-spaces will have nonnegative CQB and positive $^d$CQB. By giving irreducible such examples with arbitrarily large second Betti numbers we show that the positivity of these two curvature put no restriction on the Betti number. A strengthened conjecture is that any Kahler C-space will actually have positive CQB unless it is a ${mathbb P}^1$ bundle. Finally we give an example of non-symmetric, irreducible Kahler C-space with $b_2>1$ and positive CQB, as well as compact non-locally symmetric Kahler manifolds with CQB$<0$ and $^d$CQB$<0$.
In this paper, we show that a closed $n$-dimensional generalized ($lambda, n+m)$-Einstein manifold with positive isotropic curvature and constant scalar curvature must be isometric to either a sphere ${Bbb S}^n$, or a product ${Bbb S}^{1} times {Bbb S}^{n-1}$ of a circle with an $(n-1)$-sphere, up to finite cover and rescaling.
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