A Survey of Riemannian Contact Geometry

Abstract

This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

The Einstein-Hilbert Type Action on Pseudo-Riemannian Almost-Product Manifolds

“…On a manifold equipped with an additional structure (e.g., almost product, complex or contact), one can consider an analogue of (2) adjusted to that structure. In pseudo-Riemannian geometry, it may mean restricting g to a certain class of metrics (e.g., conformal to a given one, in the Yamabe problem [6]) or even constructing a new, related action (e.g., the Futaki functional on a Kahler manifold [6], or several actions on contact manifolds [7]), to cite only few examples. The latter approach was taken in authors' previous papers, where the scalar curvature in the Einstein-Hilbert action on a pseudo-Riemannian manifold was replaced by the mixed scalar curvature of a given distribution or a foliation.…”

“…In Section 3.1, we vary functional (7) with respect to metric g. Compared to its variation with fixed g, which was considered in [25], we obtain additional conditions for general and metric connections. On the other hand, a metric-affine doubly twisted product is critical for (7) if and only if it is critical for the action with fixed g. Similarly, restricting (7) to pairs of metrics and statistical connections also does not give any new Euler-Lagrange equations than those obtained in [25].…”

The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds, II

“…On a manifold equipped with an additional structure (e.g., almost product, complex or contact), one can consider an analogue of (2) adjusted to that structure. In pseudo-Riemannian geometry, it may mean restricting g to a certain class of metrics (e.g., conformal to a given one, in the Yamabe problem [7]) or even constructing a new, related action (e.g., the Futaki functional on a Kahler manifold [7], or several actions on contact manifolds [8]), to cite only few examples. The latter approach was taken in authors' previous papers, where the scalar curvature in the Einstein-Hilbert action on a pseudo-Riemannian manifold was replaced by the mixed scalar curvature of a given distribution or a foliation.…”

“…Example 3. In [13] it was proved that on a Sasaki manifold (M, g, ξ, η) (that is, M with a normal contact metric structure [8]) there exists a unique metric connection with a skew-symmetric, parallel torsion tensor, and its contorsion tensor is given by T X Y, Z = 1 2 (η ∧ dη)(X, Y, Z), where X, Y, Z ∈ X M and η is the contact form on M . Let D be the one-dimensional distribution spanned by the Reeb field ξ.…”

“…), see (46), as dη(X, Y ) = 2 X,T ξ Y (we use here the same convention for differential of forms as in [13], which is different than the one in [8]). Since g is a Sasaki metric, both distributions are totally geodesic, and for volumepreserving variations the Euler-Lagrange equation (47a) gets λ g ⊥ on the righthand side (see Remark 3) and becomes…”