axioms Article Remarkable Classes of Almost 3-Contact Metric Manifolds Giulia Dileo Department of Mathematics, University of Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy; [email protected] Abstract: We introduce a new class of almost 3-contact metric manifolds, called 3-(0, d)-Sasaki manifolds. We show fundamental geometric properties of these manifolds, analyzing analogies and differences with the known classes of 3-(a, d)-Sasaki (a 6= 0) and 3-d-cosymplectic manifolds. Keywords: almost 3-contact metric manifold; 3-Sasaki; 3-cosymplectic; 3-(a, d)-Sasaki; 3-d- cosym- plectic; metric connection with skew torsion MSC: 53C15; 53C25; 53B05 1. Introduction An almost 3-contact metric manifold is a (4n + 3)-dimensional differentiable manifold M endowed with three almost contact metric structures (j,xi, hi, g), i = 1, 2, 3, sharing the same Riemannian metric g and satisfying suitable compatibility conditions, equivalent to the existence of a sphere of almost contact metric structures. In the recent paper [1], new classes of almost 3-contact metric manifolds were introduced and studied. The first remarkable class is given by 3-(a, d)-Sasaki manifolds defined as almost 3-contact metric manifolds (M, ji, xi, hi, g) such that ∗ dhi = 2aFi + 2(a − d)hj ^ hk, a 2 R , d 2 R, (1) Citation: Dileo, G. Remarkable for every even permutation (i, j, k) of (1, 2, 3). This is a generalization of 3-Sasaki manifolds, Classes of Almost 3-Contact Metric which correspond to the values a = d = 1. A second class introduced in [1] is given by Manifolds. Axioms 2021, 10, 8. https://doi.org/10.3390/axioms 3-d-cosymplectic manifolds defined by the conditions 10010008 dhi = −2dhj ^ hk, dFi = 0, d 2 R, Received: 9 December 2020 generalizing 3-cosymplectic manifolds which correspond to the value d = 0. Accepted: 7 January 2021 In the present paper we will introduce a third class of almost 3-contact metric mani- Published: 11 January 2021 folds, which is in fact a second (and alternative) generalization of 3-cosymplectic manifolds. Publisher’s Note: MDPI stays neu- We will consider almost 3-contact metric manifolds whose structure tensor fields satisfy tral with regard to jurisdictional clai- dh = −2dh ^ h , dF = −2d(h ^ F − h ^ F ), d 2 (2) ms in published maps and institutio- i j k i j k k j R nal affiliations. for every even permutation (i, j, k) of (1, 2, 3). When d = 0 we recover a 3-cosymplectic manifold. We will call these manifolds 3-(0, d)-Sasaki manifolds. The choice of name is due to the fact that for a 3-(a, d)-Sasaki manifold, Equation (1) implies Copyright: © 2021 by the author. Li- censee MDPI, Basel, Switzerland. dFi = 2(a − d)(hj ^ Fk − hk ^ Fj), (3) This article is an open access article distributed under the terms and con- so that the two equations in (2) formally correspond to (1) and (3) with a = 0, although ditions of the Creative Commons At- in this case the second equation is no more a consequence of the first one. In fact the two tribution (CC BY) license (https:// conditions in (2) are not completely independent (see Remark1). Examples of 3-(0, d)- 4n creativecommons.org/licenses/by/ Sasaki structures can be defined on the semidirect products SO(3) n R . The structure on 4.0/). these Lie groups was introduced in [2] as an example of canonical abelian almost 3-contact Axioms 2021, 10, 8. https://doi.org/10.3390/axioms10010008 https://www.mdpi.com/journal/axioms Axioms 2021, 10, 8 2 of 14 metric structure. It is also shown in [2] that the Lie group SO(3) n R4n admits co-compact discrete subgroups, so that the corresponding compact quotients admit almost 3-contact metric structures of the same type. One can show that for all the above three classes of manifolds, 3-(a, d)-Sasaki, 3-d- cosymplectic, and 3-(0, d)-Sasaki manifolds, the structure is hypernormal, the characteristic vector fields xi, i = 1, 2, 3, are Killing and they span an integrable distribution, called vertical, with totally geodesic leaves. Nevertheless, there are remarkable geometric differ- ences between the three classes. In the 3-(a, d)-Sasaki case the 1-forms hi are all contact n forms, i.e., hi ^ (dhi) 6= 0 everywhere on M, while for the other two classes, the hor- izontal distribution defined by hi = 0, i = 1, 2, 3, is integrable. Both 3-d-cosymplectic manifolds and 3-(0, d)-Sasaki manifolds are locally isometric to the Riemannian product of a 3-dimensional Lie group, tangent to the vertical distribution, and a 4n-dimensional manifold tangent to the horizontal distribution. The Lie group is either isomorphic to SO(3) or flat depending on wether d 6= 0 or d = 0. Each horizontal leaf is endowed with a hyper-Kähler structure. The difference between 3-d-cosymplectic and 3-(0, d)-Sasaki manifolds lies in the projectability of the structure tensor fields ji, i = 1, 2, 3, with respect to the vertical foliation. They are always projectable for 3-d-cosymplectic manifolds, but not for 3-(0, d)-Sasaki manifolds with d 6= 0. In this case one can project a transverse quaternionic structure, as it happens for 3-(a, d)-Sasaki manifolds. Finally, for the three classes of manifolds, we analyze the existence of a canonical metric connection with totally skew-symmetric torsion. 2. Almost Contact and Almost 3-Contact Metric Manifolds An almost contact manifold is a smooth manifold M of dimension 2n + 1, endowed with a structure (j, x, h), where j is a (1, 1)-tensor field, x a vector field, and h a 1-form such that j2 = −I + h ⊗ x, h(x) = 1, implying that jx = 0, h ◦ j = 0, and j has rank 2n. The tangent bundle of M splits as TM = H ⊕ hxi, where H is the 2n-dimensional distribution defined by H = Im(j) = Ker(h). The vector field x is called the characteristic or Reeb vector field. On the product manifold M × R one can define an almost complex structure J by d d J X, f dt = jX − f x, h(X) dt , where X is a vector field tangent to M, t is the coordinate of R and f is a C¥ function on M × R. If J is integrable, the almost contact structure is said to be normal and this is equivalent to the vanishing of the tensor field Nj := [j, j] + dh ⊗ x, where [j, j] is the Nijenhuis torsion of j [3]. More precisely, for any vector fields X and Y, Nj is given by 2 Nj(X, Y) = [jX, jY] + j [X, Y] − j[jX, Y] − j[X, jY] + dh(X, Y)x. (4) It is known that any almost contact manifold admits a compatible metric, that is a Riemannian metric g such that g(jX, jY) = g(X, Y) − h(X)h(Y) for every X, Y 2 X(M). Then h = g(·, x) and H = hxi?. The manifold (M, j, x, h, g) is called an almost contact metric manifold. The associated fundamental 2-form is defined by F(X, Y) = g(X, jY). We recall some remarkable classes of almost contact metric manifolds. • An a-contact metric manifold is defined as an almost contact metric manifold such that ∗ dh = 2aF, a 2 R , When a = 1, it is called a contact metric manifold; the 1-form h is a contact form, that is h ^ (dh)n 6= 0 everywhere on M. An a-Sasaki manifold is a normal a-contact metric manifold, and again such a manifold with a = 1 is called a Sasaki manifold. • An almost cosymplectic manifold is defined as an almost contact metric manifold such that dh = 0, dF = 0; Axioms 2021, 10, 8 3 of 14 if furhermore the structure is normal, M is called a cosymplectic manifold. It is worth remarking that some authors call these manifolds almost coKähler and coKähler, respec- tively ([4]). • A quasi-Sasaki manifold is a normal almost contact metric manifold with closed 2-form F. This class includes both a-Sasaki and cosymplectic manifolds. The Reeb vector field of a quasi-Sasaki manifold is always Killing. Both a-Sasaki manifolds and cosymplectic manifolds can be characterized by means of the Levi-Civita connection rg. Indeed, one can show that an almost contact metric manifold (M, j, x, h, g) is a-Sasaki if and only if g (rX j)Y = a(g(X, Y)x − h(X)Y) 8X, Y 2 X(M). An almost contact metric manifold is cosymplectic if and only if rg j = 0; further, this is equivalent to requiring the manifold to be locally isometric to the Riemannian product of a real line (tangent to the Reeb vector field) and a Kähler manifold. For a comprehensive introduction to almost contact metric manifolds we refer to [3]. For Sasaki geometry, we also recommend the monograph [5]; the survey [4] covers funda- mental properties and recent results on cosymplectic geometry. An almost 3-contact manifold is a differentiable manifold M of dimension 4n + 3 en- dowed with three almost contact structures (ji, xi, hi), i = 1, 2, 3, satisfying the following relations, jk = ji jj − hj ⊗ xi = −jj ji + hi ⊗ xj, xk = jixj = −jjxi, hk = hi ◦ jj = −hj ◦ ji, for any even permutation (i, j, k) of (1, 2, 3) ([3]). The tangent bundle of M splits as TM = H ⊕ V, where 3 \ H := Ker(hi), V := hx1, x2, x3i. i=1 In particular, H has rank 4n.