[Paper Review] On the geometry of the connection with totally skew-symmetric torsion on almost complex manifolds with Norden metric

Dimitar Mekerov|arXiv (Cornell University)|Feb 5, 2009

Geometric Analysis and Curvature Flows7 references6 citations

TL;DR

This paper investigates a unique linear connection with totally skew-symmetric torsion on almost complex manifolds equipped with a Norden metric, proving its existence and uniqueness on non-Kähler manifolds implies quasi-Kähler structure. The key contribution is establishing that such a connection exists if and only if the manifold is quasi-Kählerian, and it derives explicit formulas for the connection and curvature tensor, showing that parallel torsion and Kähler curvature imply isotropic Kähler structure.

ABSTRACT

We consider an almost complex manifold with Norden metric (i. e. a metric with respect to which the almost complex structure is an anti-isometry). On such a manifold we study a linear connection preserving the almost complex structure and the metric and having a totally skew symmetric torsion tensor (i. e. a 3-form). We prove that if a non-Kaehler almost complex manifold with Norden metric admits such connection then the manifold is quasi-Kaehlerian (i. e. has non-integrable almost complex structure). We prove that this connection is unique, determine its form, and construct an example of it on a Lie group. We consider the case when the manifold admits a connection with parallel totally skew-symmetric torsion and the case when such connection has a Kaehler curvature tensor. We get necessary and sufficient conditions for an isotropic Kaehler manifold with Norden metric.

Motivation & Objective

• To establish the existence and uniqueness of a linear connection with totally skew-symmetric torsion on almost complex manifolds with Norden metric.
• To characterize the geometric conditions under which such a connection exists, particularly in non-Kähler settings.
• To determine the form of the connection and its curvature tensor, especially when torsion is parallel or curvature is Kählerian.
• To identify necessary and sufficient conditions for the manifold to be isotropic Kählerian in terms of scalar curvatures and curvature tensors.

Proposed method

• Constructs a linear connection ∇′ preserving the almost complex structure J and the Norden metric g, with torsion tensor T being a 3-form (totally skew-symmetric).
• Derives the explicit form of the connection ∇′ via the transformation tensor Q, relating it to the Levi-Civita connection ∇ and the Nijenhuis tensor via ∇J.
• Uses the curvature formula R′(x,y,z,w) = R(x,y,z,w) + 2g(Q(x,y),Q(z,w)) − g(Q(x,z),Q(y,w)) + g(Q(y,z),Q(x,w)) to analyze curvature properties.
• Applies the first Bianchi identity to determine conditions under which R′ is a Kähler tensor, leading to the condition ∑S g(Q(x,y),Q(z,w)) = 0.
• Employs the square norm ||∇J||² to define isotropic Kähler manifolds and relates scalar curvatures of ∇ and ∇′.
• Constructs a 4-parametric family of 4-dimensional quasi-Kähler manifolds via a Lie group to provide a concrete example of the connection.

Experimental results

Research questions

• RQ1 Under what conditions does a non-Kähler almost complex manifold with Norden metric admit a linear connection with totally skew-symmetric torsion preserving J and g?
• RQ2 What is the explicit form of such a connection, and is it unique?
• RQ3 When does the torsion of this connection remain parallel, and what geometric constraints does this impose?
• RQ4 When is the curvature tensor of this connection Kählerian, and what does this imply about the underlying manifold?
• RQ5 What is the relationship between the scalar curvatures of the Levi-Civita and the new connection, and when are they equal?

Key findings

• A non-Kähler almost complex manifold with Norden metric admits a linear connection with totally skew-symmetric torsion if and only if it is quasi-Kählerian.
• The connection ∇′ is unique and its form is explicitly derived in terms of the Levi-Civita connection and the tensor ∇J.
• The connection ∇′ has a Kähler curvature tensor if and only if the cyclic sum ∑S g(Q(x,y),Q(z,w)) = 0 holds.
• If ∇′ has both parallel torsion and Kähler curvature tensor, then the manifold is isotropic Kählerian, i.e., ||∇J||² = 0.
• The scalar curvatures of ∇ and ∇′ are equal if and only if the manifold is isotropic Kählerian.
• A 4-parametric family of 4-dimensional quasi-Kähler manifolds with Norden metric is constructed on a Lie group, and ∇′ is computed explicitly for this family.