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# 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

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Andrei Moroianu, Jun 12, 2015 Available from: Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

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**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. -
##### Article: Deformation theory of G_2 conifolds

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**ABSTRACT:**We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) G_2 manifolds. In the AC case, we show that if the rate of convergence nu to the cone at infinity is generic in a precise sense and lies in the interval (-4, -5/2), then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates nu < -4 in the AC case, and for generic positive rates of convergence to the cones at the singular points in the CS case, the deformation theory is in general obstructed. We describe the obstruction spaces explicitly in terms of the spectrum of the Laplacian on the link of the cones on the ends, and compute the virtual dimension of the moduli space. We also present several applications of these results, including: the local uniqueness of the Bryant--Salamon AC G_2 manifolds; the smoothness of the CS moduli space if the singularities are modeled on particular G_2 cones; and the proof of existence of a "good gauge" needed for desingularization of CS G_2 manifolds. Finally, we discuss some open problems for future study. - [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