[Paper Review] Nearly Sasakian geometry and $SU(2)$-structures

Beniamino Cappelletti–Montano, Giulia Dileo|arXiv (Cornell University)|Oct 3, 2014

Geometry and complex manifolds20 references4 citations

TL;DR

This paper establishes a systematic study of nearly Sasakian manifolds, proving they are contact manifolds and admitting integrable distributions with Sasakian or 5-dimensional nearly Sasakian leaves. It reveals a one-to-one correspondence between 5-dimensional nearly Sasakian structures and a special class of nearly hypo $SU(2)$-structures, which deform to Sasaki-Einstein structures, and introduces a canonical connection that parallelizes both the $SU(2)$-structure and torsion tensor in dimension 5.

ABSTRACT

We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian or $5$-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact manifold. Focusing on the $5$-dimensional case, we prove that there exists a one-to-one correspondence between nearly Sasakian structures and a special class of nearly hypo $SU(2)$-structures. By deforming such a $SU(2)$-structure one obtains in fact a Sasaki-Einstein structure. Further we prove that both nearly Sasakian and Sasaki-Einstein $5$-manifolds are endowed with supplementary nearly cosymplectic structures. We show that there is a one-to-one correspondence between nearly cosymplectic structures and a special class of hypo $SU(2)$-structures which is again strictly related to Sasaki-Einstein structures. Furthermore, we study the orientable hypersurfaces of a nearly Kähler 6-manifold and, in the last part of the paper, we define canonical connections for nearly Sasakian manifolds, which play a role similar to the Gray connection in the context of nearly Kähler geometry. In dimension $5$ we determine a connection which parallelizes all the nearly Sasakian $SU(2)$-structure as well as the torsion tensor field. An analogous result holds also for Sasaki-Einstein structures.

Motivation & Objective

• To systematically study nearly Sasakian manifolds and their geometric structure.
• To establish that nearly Sasakian manifolds are contact manifolds via integrable distributions with totally geodesic leaves.
• To prove a one-to-one correspondence between 5-dimensional nearly Sasakian structures and a special class of nearly hypo $SU(2)$-structures.
• To show that deforming such $SU(2)$-structures yields Sasaki-Einstein structures.
• To define and characterize a canonical connection for nearly Sasakian manifolds that parallelizes the $SU(2)$-structure and torsion tensor in dimension 5.

Proposed method

• The paper uses the tensor field $ h = - abla_ ho - \phi $ to measure non-Sasakianity and analyzes its eigenvalues and eigendistributions.
• It proves that the tangent bundle splits orthogonally into eigendistributions of $ h^2 $, with eigenvalues $ 0, -\lambda_1^2, \dots, -\lambda_r^2 $, leading to integrable, totally geodesic foliations.
• The authors establish a one-to-one correspondence between 5-dimensional nearly Sasakian structures and nearly hypo $SU(2)$-structures via the $SU(2)$-structure defined by $ (\eta, \omega_1, \omega_2, \omega_3) $.
• A canonical connection $ \bar{\nabla} $ is defined using the torsion tensor $ H $, with components involving $ \eta $, $ \phi $, $ h $, and the metric, ensuring parallelization of the $ SU(2) $-structure.
• The deformation of nearly hypo $SU(2)$-structures to Sasaki-Einstein structures is achieved via a specific conformal change of the metric.
• The canonical connection is shown to preserve both the $ SU(2) $-structure and the torsion tensor, particularly when $ r = \frac{1}{2} $.

Experimental results

Research questions

• RQ1 How do nearly Sasakian manifolds relate to Sasakian and Sasaki-Einstein structures through $ SU(2) $-structures?
• RQ2 What is the role of the tensor field $ h $ in characterizing the geometric decomposition of nearly Sasakian manifolds?
• RQ3 Can a canonical connection be defined for nearly Sasakian manifolds that parallelizes the $ SU(2) $-structure and torsion tensor?
• RQ4 Is there a one-to-one correspondence between 5-dimensional nearly Sasakian structures and a special class of nearly hypo $ SU(2) $-structures?
• RQ5 How does deformation of nearly hypo $ SU(2) $-structures lead to Sasaki-Einstein structures?

Key findings

• Any nearly Sasakian manifold admits an orthogonal decomposition of its tangent bundle into eigendistributions of $ h^2 $, with eigenvalues $ 0, -\lambda_1^2, \dots, -\lambda_r^2 $, leading to integrable, totally geodesic foliations.
• The distribution $ \mathcal{D}(0) $ is integrable and its leaves are Sasakian manifolds of dimension $ 2p+1 $, with $ p > 0 $ implying non-trivial Sasakian structure.
• Each distribution $ [\xi] \oplus \mathcal{D}(-\lambda_i^2) $ is integrable and defines a totally geodesic foliation with 5-dimensional nearly Sasakian non-Sasakian leaves.
• The leaf space of the foliation $ [\xi] \oplus \mathcal{D}(-\lambda_1^2) \oplus \cdots \oplus \mathcal{D}(-\lambda_r^2) $ is Kähler when $ p > 0 $, establishing a Kähler reduction.
• There exists a one-to-one correspondence between 5-dimensional nearly Sasakian structures and a special class of nearly hypo $ SU(2) $-structures, which deform to Sasaki-Einstein structures.
• A canonical connection $ \bar{\nabla} $ exists in dimension 5 that parallelizes both the $ SU(2) $-structure and the torsion tensor, particularly when $ r = \frac{1}{2} $, and coincides with the Okumura connection for Sasakian structures.