On characteristic foliations of metric contact-symplectic structures

We study compatible and associated metrics for a contact-symplectic pair $(η, ω)$ on a manifold. We show that the integral curves of the Reeb vector field are geodesics for any compatible metric. We prove that all associated metrics share a common volume element, which we give explicitly. When the characteristic foliations of $η$ and $ω$ are orthogonal with respect to an associated metric, their leaves, as well as those of the characteristic foliation of $dη$, are minimal. We construct explicit examples on nilpotent Lie groups and nilmanifolds where the characteristic foliations are not both totally geodesic.

💡 Research Summary

The paper investigates metric aspects of contact‑symplectic pairs ((\eta,\omega)) on odd‑dimensional manifolds. A contact‑symplectic pair consists of a 1‑form (\eta) and a closed 2‑form (\omega) satisfying (\eta\wedge(d\eta)^{m}\wedge\omega^{n}\neq0), ((d\eta)^{m+1}=0) and (\omega^{,n+1}=0) for some integers (m,n\ge1). Under these conditions the manifold splits into two complementary, transverse foliations: the characteristic foliation (S) of (\eta) (dimension (2n)) whose leaves inherit the symplectic form (\omega), and the characteristic foliation (C) of (\omega) (dimension (2m+1)) whose leaves inherit the contact form (\eta). The Reeb vector field (\xi) is uniquely defined by (\eta(\xi)=1,\ i_{\xi}d\eta=0,\ i_{\xi}\omega=0); it is tangent to (C) and restricts to the usual Reeb field on each contact leaf. Its flow preserves both (\eta) and (\omega).

The authors introduce an auxiliary ((1,1))-tensor (\varphi) satisfying (\varphi^{2}=-I+\eta\otimes\xi). This makes ((\eta,\omega,\varphi)) an almost contact‑symplectic structure: (\varphi) vanishes on (\xi), annihilates (\eta), and induces an almost complex structure on the horizontal distribution (H=\ker\eta). Two types of Riemannian metrics are then defined:

• Compatible metrics satisfy (g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y)). Such a metric automatically makes (\xi) unit length and orthogonal to (H).

• Associated metrics are compatible and additionally satisfy (g(X,\varphi Y)=(d\eta+\omega)(X,Y)) together with (g(X,\xi)=\eta(X)). Every almost contact‑symplectic structure admits compatible metrics, and any associated metric is automatically compatible, but the converse fails in general (the paper shows this by considering the pair ((\eta,-\omega,\varphi))).

Theorem 7 proves that for any compatible metric the integral curves of the Reeb vector field are geodesics. The proof uses the invariance (\mathcal L_{\xi}\eta=0) and the unit‑length condition (g(\xi,\xi)=1) to obtain (\nabla_{\xi}\xi=0). Hence the Reeb flow is a geodesic flow for all compatible metrics, extending a well‑known fact from pure contact geometry.

Theorem 8 gives a universal volume form for all associated metrics: \