In 2002, Frédéric Bourgeois [2] showed that, given a compact contact manifold M 2n–1, the product M 2n–1 × T 2 also carries a contact form; then, as a corollary he observed that all odd-dimensional tori have contact forms. The idea is to use an open book decomposition of M 2n–1 that is compatible with its contact structure [4] to produce contact forms on the product M 2n–1 × T 2. Here we first find the Reeb vector field and then a method for constructing associated metrics for contact forms of the type studied by Bourgeois. We will also discuss S 2 × T 2 in some detail. While the procedure would apply to tori, the construction is quite difficult and we will make only some remarks.

In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field.

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.

We consider g-natural metrics on the tangent bundle of a Riemannian manifold together with the almost complex structure which reverses the horizontal and vertical subspaces. This narrows the class of g-natural metrics to metrics conformally equivalent to the Sasaki metric on the tangent bundle with a restriction on the conformal factor. We then show that such a g-natural almost Hermitian structure is Bochner flat if and only if it is conformally equivalent to the Sasaki metric when the base manifold is flat and with the same restriction on the conformal factor.

In this paper, we first focus on conformally flat almost \({C(\alpha)}\)-manifolds. Moreover, we construct an example of a 3-dimensional conformally flat almost \({\alpha}\)-Kenmotsu manifold which is of non-constant sectional curvature. By means of this example, we also illustrate a 3-dimensional conformally flat almost Kenmotsu manifold which not only contrasts with both H 3(−1) and \({H^{2}(-4)\times\mathbb{R}}\) but also is of non-constant sectional curvature. Then, we study conformally flat, \({\phi}\)-RK contact metric and almost contact metric manifolds. Next, we investigate the curvature properties of conformally flat generalized Sasakian space forms. Finally, we deal with conformally flat almost contact metric manifolds which are *-\({\eta}\)-Einstein. In this regard, we are also interested in conformally flat contact metric manifolds of \({{\rm dim}{\geq}5}\) which are *-\({\eta}\)-Einstein.

We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner-flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-Kähler Vaisman manifolds.

Treatment of filamentous fungal infections relies on a limited repertoire of antifungal agents. Compounds possessing novel modes of action are urgently required. N-myristoylation is a ubiquitous modification of eukaryotic proteins. The enzyme N-myristoyltransferase (NMT) has been considered a potential therapeutic target in protozoa and yeasts. Here, we show that the filamentous fungal pathogen Aspergillus fumigatus possesses an active NMT enzyme that is essential for survival. Surprisingly, partial repression of the gene revealed downstream effects of N-myristoylation on cell wall morphology. Screening a library of inhibitors led to the discovery of a pyrazole sulphonamide compound that inhibits the enzyme and is fungicidal under partially repressive nmt conditions. Together with a crystallographic complex showing the inhibitor binding in the peptide substrate pocket, we provide evidence of NMT being a potential drug target in A. fumigatus.

In this lecture we will discuss various generalizations of the catenoid and the helicoid as well as related differential geometric notions including minimality, quasi-umbilicity and conformal flatness.

We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold with decomposable ϕ is a Calabi‐Eckmann manifold or the Riemannian product of a sphere and . We show that a complete, simply connected, normal metric contact pair manifold with decomposable ϕ, such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, is the product of globally ϕ‐symmetric spaces or the product of a globally ϕ‐symmetric space and . Moreover in the first case the manifold fibers over a locally symmetric space endowed with a symplectic pair.

We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the sum of the two Reeb vector fields. This shows that metrics associated to normal contact pairs cannot be flat. Therefore flat non-Kähler Vaisman manifolds do not exist. Furthermore we give a local classification of metric contact pair manifolds whose curvature vanishes on the vertical subbundle. As a corollary we have that flat associated metrics can only exist if the leaves of the characteristic foliations are at most three-dimensional.

The goal of this lecture will be to introduce the notion ofD-homothetic warping, give a few rudimentary properties and a couple of applications. As with the usual warped product it is hoped that this idea will prove useful for generating further results and examples of various structures. Details of the proofs will appear in [3]. For this purpose we must first review the geometry of contact metric and almost contact metric manifolds. By a contact manifold we mean a C manifold M2n+1 together with a 1-form η such that η ∧ (dη) 6= 0. It is well known that given η there exists a unique vector field ξ such that dη(ξ,X) = 0 and η(ξ) = 1. The vector field ξ is known as the characteristic vector field or Reeb vector field of the contact structure η. Denote by D the contact subbundle defined by {X ∈ TmM : η(X) = 0}. A Riemannian metric g is an associated metric for a contact form η if, first of all, η(X) = g(X, ξ) and secondly, there exists a field of endomorphisms, φ, such that φ2 = −I + η ⊗ ξ, dη(X,Y ) = g(X,φY ). We refer to (φ, ξ, η, g) as a contact metric structure and to M2n+1 with such a structure as a contact metric manifold. By an almost contact manifold we mean a C manifold M2n+1 together with a field of endomorphisms φ, a 1-form η and a vector field ξ such that

We prove that a complex ( κ , µ)-space with κ < 1 is a locally homogeneous complex contact metric manifold. Also, a complex ( κ , µ)-space has either κ = 1 or is GH -locally symmetric.

We first show that a locally symmetric normal complex contact metric manifold is locally isometric to the complex projective space with the standard Fubini-Study metric. We then study reflections in the integral submanifolds of the vertical subbundle of a regular normal complex contact metric manifold. If the reflections are isometries, the manifold fibers over a locally symmetric space. Moreover, if the normal complex contact metric manifold is Kahler, then the manifold fibers over a quaternionic symmetric space. On the other hand, if the complex contact structure is given by a global holomorphic contact form, then the manifold fibers over a locally symmetric complex symplectic manifold.

Generalized Sasakian-space-forms are introduced and studied. Many examples of these manifolds are presented, by using some different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations. New results on generalized complex-space-forms are also obtained.

In this article, we study discrete curvature and torsion for spatial polygonal lines of unit sides. We express geometric conditions on polygons by using inner products of oriented sides. As a application, we prove a generalization of van der Waerden's theorem. The theorem given in this paper and its proof clarify how the conditions on the sides affect the polygon being planar from the discrete torsion point of view.

Before turning to our main topics we first discuss partially hyperbolic diffeomorphisms and holomorphic Anosov flows as introduced by Étienne Ghys [1995]. In Section 13.2 we discuss the geometry of the projectivized holomorphic tangent and cotangent bundles. The study of the projectivized holomorphic tangent bundle naturally raises the question of a complex geodesic flow, which we discuss in Section 13.3. In Section 13.4 we return to the projectivized holomorphic tangent bundle and develop its complex almost contact metric structure. In Section 13.5 we first discuss special directions on complex contact manifolds analogous to our treatment in the real case in Chapter 11 and then discuss complex contact structures on the Lie group SL(2, $$ C\!\!\!\!C $$ ) in detail.

The study of the integral of the scalar curvature, A(g) = ∫ M τ dV g , as a functional on the set M 1 of all Riemannian metrics of the same total volume on a compact orientable manifold M is now classical, dating back to Hilbert [1915] (see also Nagano [1967]). A Riemannian metric g is a critical point of A(g) if and only if g is an Einstein metric. Since there are so many Riemannian metrics on a manifold, one can regard, philosophically, the finding of critical metrics as an approach to searching for the best metric for the given manifold. Other functions of the curvature have been taken as integrands as well, most notably % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaacI % cacaWGNbGaaiykaiabg2da9maapebabaGaeqiXdq3aaWbaaSqabeaa % caaIYaaaaOGaamizaiaadAfadaWgaaWcbaGaam4zaaqabaGccaGGSa % GaaGjbVlaadoeacaGGOaGaam4zaiaacMcacqGH9aqpdaWdraqaamaa % emaabaGaeqyWdihacaGLhWUaayjcSdaaleaacaWGnbaabeqdcqGHRi % I8aaWcbaGaamytaaqab0Gaey4kIipakmaaCaaaleqabaGaaGOmaaaa % kiaadsgacaWGwbWaaSbaaSqaaiaadEgaaeqaaaaa!5465! $$ B(g) = {\int_M {{\tau ^2}d{V_g},\;C(g) = \int_M {\left| \rho \right|} } ^2}d{V_g} $$ where ρ is the Ricci tensor, and % MathType!MTEF!2!1!+- % feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiaacI % cacaWGNbGaaiykaiabg2da9maapebabaWaaqWaaeaacaWGsbWaaSba % aSqaaiaadUgacaWGQbGaamyAaiaadIgaaeqaaaGccaGLhWUaayjcSd % WaaWbaaSqabeaacaaIYaaaaaqaaiaad2eaaeqaniabgUIiYdGccaWG % KbGaamOvamaaBaaaleaacaWGNbaabeaaaaa!4890! $$D(g) = \int_M {{{\left| {{R_{kjih}}} \right|}^2}} d{V_g}$$ ; the critical point conditions for these have been computed by Berger [1970]. From the critical point conditions it is easy to see that Einstein metrics are critical for B(g) and C(g) but not necessarily conversely. For example an η-Einstein manifold M 2n+1 with scalar curvature equal to 2n(2n + 1) or 2n(2n + 3) is a non-Einstein critical metric of C(g), Yamaguchi and Chūman [1983]. In the case of B(g) Yamaguchi and Chūman showed that a Sasakian critical point is Einstein. Similarly metrics of constant curvature and Kähler metrics of constant holomorphic curvature are critical for D(g), see Muto [1975]; also a Sasakian manifold of dimension m and constant ϕ-sectional curvature 3m — 1 is critical for D(g), see Yamaguchi and Chūman [1983].

In the first two sections of this chapter we discuss the geometry of the tangent bundle and the tangent sphere bundle. In Section 3 we briefly present a more general construction on vector bundles and in Section 4 specialize to the case of the normal bundle of a submanifold. The formalism for the tangent bundle and the tangent sphere bundle is of sufficient importance to warrant its own development, rather than specializing from the vector bundle case. As we saw in Chapter 1, the cotangent bundle of a manifold has a natural symplectic structure and we will see here that the same is true of the tangent bundle of a Riemannian manifold.