The Hermitian Laplace Operator on Nearly Kähler Manifolds

Communications in Mathematical Physics (Impact Factor: 1.9). 01/2010; DOI: 10.1007/s00220-009-0903-4
Source: OAI

ABSTRACT The moduli space NK of infinitesimal deformations of a nearly Kähler 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 Kähler manifolds. It turns out that the nearly Kähler 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 Kähler deformations, modeled on the Lie algebra su_3 of the isometry group.

  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: For a compact connected Lie group $G$ we prove that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra $\frak{g}$ coincide with the bi-invariant metric connections. In the sequel, we focus on the geometry of a naturally reductive space $(M=G/K, g)$ endowed with a family of $G$-invariant connections $\{\nabla^{\alpha} : \alpha\in\mathbb{R}\}$ whose torsion is a multiple of the torsion of the canonical connection $\nabla^{c}$, i.e. $T^{\alpha}=\alpha\cdot T^{c}$. For the spheres $S^{6}$ and $S^{7}$ we prove that the space of $G_2$ (resp. $Spin(7)$)-invariant affine or metric connections consists of the family $\nabla^{\alpha}$. In the compact case we examine the flatness condition $R^{\alpha}\equiv 0$ and we state a refinement of the classical Cartan-Schouten theorem. The "constancy" of the induced Ricci tensor $Ric^{\alpha}$ is also described. We prove that any compact isotropy irreducible naturally reductive Riemannian manifold, which is not a symmetric space of Type I, carries at least two $\nabla^{\alpha}$-Einstein structures with skew-torsion, namely these which occur for $\alpha=\pm 1$. A generalization of this result is given also for a class of compact normal homogeneous spaces $M=G/K$ with two isotropy summands. We introduce a new 2-parameter family of $G$-invariant connections on $M=G/K$, namely $\nabla^{s, t}$ with $s\in\mathbb{R}$ and $t\in\mathbb{R}_{+}$; for the Killing metric $t=1/2$ skew-torsion appears and we examine the $\nabla^{s, 1/2}$-Einstein condition. We show that $M$ is normal Einstein, if and only if, $M$ is a $\nabla^{s, 1/2}$-Einstein manifold with skew-torsion for one of the values $s=0, 2$. In this way we provide a series of new examples of manifolds admitting these structures.admitting these structures.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We study generalized Killing spinors on the standard sphere $\mathbb{S}^3$, which turn out to be related to Lagrangian embeddings in the nearly K\"ahler manifold $S^3\times S^3$ and to great circle flows on $\mathbb{S}^3$. Using our methods we generalize a well known result of Gluck and Gu concerning divergence-free geodesic vector fields on the sphere and we show that the space of Lagrangian submanifolds of $S^3\times S^3$ has at least three connected components.
    Differential Geometry and its Applications 05/2014; 37. DOI:10.1016/j.difgeo.2014.09.005 · 0.59 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus the dimension of Killing spinors is not preserved under Einstein deformations but is only upper semi-continuous.

Full-text (3 Sources)

Available from
Jun 5, 2014