[Paper Review] Almost contact metric structures defined by an $N$-prolonged connection

Sergey V. Galaev|arXiv (Cornell University)|Jun 5, 2015

Fixed Point Theorems Analysis4 references3 citations

TL;DR

This paper introduces the $N$-prolonged connection as a generalization of the Sasaki construction for odd-dimensional manifolds with almost contact metric structures. By extending the interior connection via an endomorphism $N$, it defines a new almost contact metric structure on the distribution $D$, proving the existence and uniqueness of a torsion-free $N$-prolonged metric connection. The key result is that the prolonged structure is almost K-contact if and only if the original structure is K-contact, and almost normal if and only if the Schouten curvature tensor of $D$ vanishes.

ABSTRACT

On a manifold with an almost contact metric structure $(φ,\vecξ,η,g,X,D)$ the notions of the interior and the $N$-prolonged connections are introduced. Using the $N$-prolonged connection, a new almost contact metric structure is defined on the distribution $D$. The properties of this structure are studied.

Motivation & Objective

• To generalize the Sasaki construction of Riemannian metrics on tangent bundles to odd-dimensional manifolds with almost contact metric structures.
• To define a new almost contact metric structure on the total space of the distribution $D$ using an $N$-prolonged connection.
• To study the geometric properties of the prolonged structure, particularly its almost contact and K-contact characteristics.
• To investigate the conditions under which the prolonged structure inherits normality or almost normality from the base structure.
• To clarify the relationship between the $N$-prolonged connection and known connections on almost contact metric manifolds.

Proposed method

• Introduces the notion of an $N$-prolonged connection as a connection in the vector bundle $(D, o, X)$, defined by an interior connection and an endomorphism $N: D \to D$.
• Defines the $N$-connection via $\nabla^N_{\vec{X}}\vec{Y} = \nabla^B_{\vec{X}}\vec{Y} + \eta(\vec{X})N\vec{Y}$, where $\nabla^B$ is the Bejancu connection.
• Constructs a prolonged almost contact metric structure on $D$ using the $N$-prolonged connection, defining the almost complex structure $J$, the Reeb vector field $\vec{u}$, and the metric $\tilde{g}$.
• Expresses the prolonged metric $\tilde{g}$ in adapted coordinates, with $\tilde{g}(\vec{\epsilon}_a, \vec{\epsilon}_b) = g_{ab}$, $\tilde{g}(\partial_{n+a}, \partial_{n+b}) = g_{ab}$, and cross-terms zero.
• Derives the Nijenhuis torsion of $J$ in terms of the Schouten curvature tensor and the endomorphism $N$, showing $N_J(\vec{\epsilon}_a, \vec{\epsilon}_b) = -R^e_{abc}x^{n+c}\partial_{n+e}$.
• Uses the Lie derivative $L_{\vec{u}}\tilde{g}$ to analyze the K-contact property, showing $L_{\vec{u}}\tilde{g} = 0$ if and only if $\partial_n g_{ab} = 0$, $N^c_b = 0$, and $P^c_{bd} = 0$.

Experimental results

Research questions

• RQ1 Under what conditions does the $N$-prolonged connection yield a metric connection with zero torsion?
• RQ2 When is the prolonged almost contact metric structure on $D$ almost K-contact?
• RQ3 What is the relationship between the almost normality of the prolonged structure and the curvature of the base distribution $D$?
• RQ4 How does the Nijenhuis torsion of the prolonged almost complex structure $J$ relate to the Schouten curvature tensor of $D$?
• RQ5 Can the manifold $D$ be isometrically embedded into the tangent bundle $TX$ with the Sasaki metric?

Key findings

• The $N$-prolonged connection exists and is unique as a metric connection with zero torsion, given an interior connection and an endomorphism $N: D \to D$.
• The prolonged almost contact metric structure is almost K-contact if and only if the original structure is K-contact, which requires $\partial_n g_{ab} = 0$, $N^c_b = 0$, and $P^c_{bd} = 0$.
• The prolonged structure is almost normal if and only if the Schouten curvature tensor of $D$ vanishes, as shown by the vanishing of the Nijenhuis torsion $N_J$.
• The fundamental form $\tilde{\omega} = d\lambda$ of the prolonged structure satisfies $\tilde{\omega}_{ab} = \omega_{ab}$, so $\text{rk}\tilde{\omega} = \frac{n-1}{2}$, implying it is not contact.
• The prolonged structure is not isometrically embeddable into the Sasaki manifold $TX$ with the standard Sasaki metric, as shown by the non-vanishing components of $L_{\vec{u}}\tilde{g}$.
• The Nijenhuis torsion of $J$ is non-zero in general, with $N_J(\vec{\epsilon}_a, \vec{\epsilon}_b) = -R^e_{abc}x^{n+c}\partial_{n+e}$, indicating the structure is not integrable unless curvature vanishes.