On the topology of metric $f$-$K$-contact manifolds

We observe that the class of metric $f$-$K$-contact manifolds, which naturally contains that of $K$-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric $f$-$K$-contact manifolds naturally splits off an exterior algebra, and relate the closed leaves of the characteristic foliation to its basic cohomology.


Introduction
An f -structure on a smooth manifold is a (1, 1)-tensor f of constant rank, satisfying f 3 + f = 0. This notion was introduced by Yano in [23] and generalizes both the notion of almost complex and of almost contact structure.
The rank of f is always even, and if maximal, then f is either an almost complex or an almost contact structure (see [23] and Section 2 for more details). f -structures with non-maximal rank (in particular with dim ker(f ) = 2) arise naturally when studying hypersurfaces of almost contact manifolds (see ).
An analogue of Hermitian structures on almost complex manifolds and of contact metric structures on almost contact manifolds was introduced on the class of f -manifolds by Blair [5]. A metric f -contact manifold is a f -manifold (M 2n+s , f ) endowed with s vector fields ξ 1 , . . . , ξ s , s one forms η 1 , . . . , η s and a Riemannian metric g such that: for every α, β ∈ {1, . . . , s} and X, Y ∈ T M , where δ β α is the Kronecker delta. The Riemannian geometry of such manifolds was studied intensively by various authors. We recall here some aspects of metric f -contact manifolds with s ≥ 2 that are very different from the metric contact setting (i.e., when s = 1). Blair [5] showed that there are no S-manifolds (i.e., normal metric f -contact manifolds, see Section 2) M 2n+s with s ≥ 2 of constant strictly positive curvature. Moreover Dileo-Lotta [11] proved the non-existence of compact, simply connected, S-manifolds M 2n+s with s ≥ 2. Obviously the situation in the Sasakian setting (i.e., when s = 1) is different.
In Section 4 we prove a splitting theorem for the de Rham cohomology of metric f -K-contact manifolds, i.e., metric f -contact manifold M whose characteristic vector fields ξ 1 , . . . , ξ s are Killing. (For s = 1 one obtains the well-known notion of a K-contact manifold.) Here F s−1 denotes the Riemannian foliation on M determined by the Killing vector fields ξ 1 , . . . , ξ s−1 , and H * (M, F s−1 ) the associated basic cohomology. The above mentioned result from [11] is a direct consequence of this.
In Section 3 we describe a new method to construct examples of (compact) metric f -(K-)contact manifolds. Starting from any metric f -contact manifold M , we construct explicitly a metric f -contact structure on the mapping torus M φ of any automorphism φ of the metric f -contact structure on M . This construction respects the subclasses of metric f -K-contact manifolds and of S-manifolds. We remark that this behavior is quite unusual; indeed, most geometric classes of manifolds are not preserved by forming mapping tori of automorphisms, see Remark 3.2.
In Section 5 and 6 we apply results from [17] to relate the closed leaves of the characteristic foliation F given by the characteristic vector fields ξ 1 , . . . , ξ s on a metric f -K-contact manifold M to the basic cohomology H * (M, F). We generalizes results of [15] in the K-contact case. The main tool is the torus T given by the closure of the flows of the characteristic vector fields in the isometry group of M , and a T -invariant Morse-Bott function S whose critical set C is equal to the union of closed leaves of F. This function generalizes a generic component of the contact momentum map in the K-contact setting, see [22,Section 4]. We obtain: If C consists of only finitely many closed leaves of F, then dim R H * (M, F) is equal to the number of closed leaves of F.
We prove moreover the following Theorem 1.3. The characteristic foliation of a compact metric f -K-contact manifold M 2n+s has at least n + 1 closed leaves. If it has only finitely many closed leaves, then the following conditions are equivalent: • The number of closed leaves of F is n + 1.
• The basic cohomology H * (M, F) is that of CP n , i.e., • The basic cohomology H * (M, F s−1 ) is that of a 2n + 1-dimensional sphere.
• M has the real cohomology ring of S 2n+1 × T s−1 .
As a consequence we obtain that any automorphism of the K-contact structure on a K-manifold M 2n+1 which has exactly n + 1 closed orbits sends every closed Reeb orbit to itself (see Corollary 6.5).

metric f -manifolds
A f -structure on a smooth manifold M 2n+s is a (1, 1) tensor f of constant rank and such that f 3 + f = 0. Given such a structure, the tangent bundle of M splits into two complementary subbundles im(f ) and ker(f ); moreover and thus the rank of f is even, say 2n (cf. [23]). If ker(f ) is parallelizable then we fix s global vector fields ξ 1 , . . . ξ s on M which span the kernel of f . Let η 1 , . . . , η s be the 1-forms determined by Then we have: In particular for s = 0 or s = 1 we have that M 2n+s is an almost complex or respectively an almost contact manifold. If in addiction the structure tensors f, ξ α , η α satisfy the normality condition: where [f, f ] denotes the Nijenhuis torsion of f , then (M, f, ξ α , η α ) is called normal, and for s = 0 or s = 1 we have that M is a complex manifold or respectively a normal almost contact manifold.
It is well-known that a manifold M 2n+s admitting an f -structure with parallelizable kernel always admits a compatible metric, that is a Riemannian metric g satisfying for every X, Y ∈ T M . The manifold M 2n+s together with the structure tensors (f, ξ α , η α , g) as above is called a metric f -manifold, and the 2-form defined by: for every α ∈ {1, . . . , s}. If a metric f -contact manifold is normal, then it is called a S-manifold.
We observe that for s = 1, the notion of metric f -contact manifold (resp. S-manifold) coincides with the notion of contact metric manifold (resp. Sasakian manifold).
We remark that one can construct metric f -contact structures on a manifold M 2n+s starting from s one-forms η α on M 2n+s satisfying some nondegeneracy condition [12, Theorem 3.1]: Theorem 2.1. Let M be a smooth manifold of dimension 2n + s admitting s one-forms η 1 , . . . , η s such that dη 1 = · · · = dη s is a 2-form of constant rank 2n and η 1 ∧ · · · ∧ η s ∧ (dη 1 ) n = 0. This result generalizes the well known construction of contact metric structures on a odd-dimensional manifold endowed with a contact form, see for instance [6,Theorem 4.4].
In the following we recall some useful properties of metric f -contact manifolds obtained in [9] and [14]. Let (M, f, ξ α , η α , g) be a metric f -contact manifold. Then the operators where L ξα denotes the Lie derivative relative to ξ α , are self-adjoint and anticommute with f . Moreover, for every α, β ∈ {1, . . . s} and X ∈ T M we have A metric f -contact manifold whose characteristic vector fields ξ 1 , . . . , ξ s are Killing is called a metric f -K-contact manifold. The following theorem is proved in [9, Theorem 2.6].
Using (2.5) we conclude that the curvature tensor field R of M satisfies for each X, Y ∈ T M and α ∈ {1, . . . , s}, where we used [21, Proposition 8.1.3] for the first equality.

Mapping tori of metric f -K-contact manifolds
In [11,Proposition 4.1] it was shown that the product of a Sasakian manifold with an Abelian Lie group always admits the structure of an Smanifold. In this section we use the same idea to show that the classes of metric f -contact, metric f -K-contact, and S-manifolds are closed under forming the mapping torus with respect to automorphisms of the structure. We begin by describing an explicit induced structure on the product with the real line.
for each X ∈ T M and where d dt denotes the standard coordinate vector field on R. We have thatf is an f -structure on M × R, im(f ) = α kerη α = im(f ), dη 1 = · · · = dη s+1 = π * 1 ω =:ω, where π 1 : M × R → M is the projection on the first component. We haveω ∧η 1 · · · ∧η s+1 = 0. The vector fieldsξ are dual toη 1 , . . . ,η s+1 and generate the kernel off . We consider moreover the Riemannian metricḡ defined bȳ Let V be a local vector field tangent to im(f ). Observe that, for each α ∈ {1, . . . , s} Now consider two local vector fields V, W tangent to im(f ) and β ∈ {1, . . . , s}. We have: Remark 3.1. Another natural choice to construct a metric f -structure on the product manifold M × R, is to consider on M × R the product metric where π 1 and π 2 are the projections from M × R on M and R respectively, and the tensorsf ,ξ α ,η α defined by: for every X ∈ T M and α ∈ {1, . . . , s}. One can easily check that, if (f, ξ α , η α , g) is a metric f -contact structure on M , then (f ,ξ α ,η α ,ḡ) is a metric f -structure on M × R; however, since dη s+1 = 0, it is not a metric f -contact structure.
The construction above, generalized to warped products, was used in [10,Example 3.3] to produce examples of generalized S-space forms from generalized Sasakian space-forms (see [1]).
We now show that the structure constructed above descends to mapping tori of automorphisms. Recall that, for a diffeomorphism φ : M → M of a manifold M the mapping torus M φ of (M, φ) is the quotient space (M ×R)/Z, where the free and properly discontinuous Z-action on the product space M × R is given by m · (p, t) = (φ m (p), t + m). Let now (M, f, ξ α , η α , g) be a metric f -contact manifold and φ : M → M an automorphism of the metric f -structure. Observe that the diffeomorphism m ∈ Z, preserves the structure tensorsf ,ξ α ,η α ,ḡ on M × R defined above. It follows that the tensorsf ,ξ α ,η α ,ḡ on M × R descend to M φ , making it a metric f -contact manifold. We have moreover that, if (M, f, ξ α , η α , g) is a (compact) metric f -K-contact manifold (or a S-manifold), then M φ with the induced structure is a (compact) metric f -K-contact manifold (or respectively a S-manifold).
Remark 3.2. Most geometric classes of manifolds are not preserved by forming mapping tori of automorphisms. For instance, the mapping torus of a symplectomorphism of a symplectic manifold naturally is a cosymplectic manifold, and that of a holomorphic isometry of a Kähler manifold is a co-Kähler manifold (see [19,Lemmata 1 and 4]). The mapping torus of a strict contactomorphism of a contact manifold is a locally conformally symplectic manifold [4,Example 2.4], and an automorphism of a Sasakian manifold induces a Vaisman structure on the mapping torus. From this point of view, metric f -K-contact structures behave in a rather unusual way.
Let M be a compact manifold. For a foliation F on M , we will consider its basic cohomology H * (M, F), which is by definition the cohomology of the subcomplex where Ξ(F) denotes the space of vector fields tangent to F.
As the ξ α are commuting Killing vector fields, they define an s-dimensional Riemannian foliation F on M , which we call the characteristic foliation of M . We will also make use of the Riemannian foliations on M spanned by the Killing vector fields ξ 1 , . . . , ξ k , for k = 1, . . . , s, which we denote by F k . The leaf dimension of F k is k; we have F s = F, and we denote by F 0 the trivial foliation by points.
Obviously, the leaves of F k are contained in those of F k+1 , for all k = 0, . . . , s − 1.
where the connecting homomorphism δ is given by δ( Proof. This follows from a variant of the Gysin sequence for pairs of foliations, whose proof is analogous to [8, Proposition 7.2.1]: Consider, for any k = 0, . . . , s − 1, the short exact sequence of complexes where T k+1 is the torus defined as the closure of the flows of the Killing vector fields defining F k+1 , i.e., ξ 1 , . . . , ξ k+1 , in the isometry group of M . The first map in the sequence is the natural inclusion. One observes that the inclusion Ω * (M, F k ) T k+1 ⊂ Ω * (M, F k ) induces an isomorphism in cohomology. (It is shown in [20, §9.1, Theorem 1] that the averaging operator Ω * (M ) → Ω * (M ) T k+1 to Ω * (M, F k ) induces an isomorphism in cohomology, and one can restrict this operator to Ω * (M, F k ). In a slightly different context, this argument was used also in [3,Lemma 5.3]).
To understand the connecting homomorphism in the induced long exact sequence in cohomology one notes that for a given closed σ ∈ Ω p−1 (M, F k+1 ) a preimage under i ξ k+1 :

Morse theory on metric f -K-contact manifolds
In this section we construct, on any compact metric f -K-contact manifold, a Morse-Bott function whose critical set is the union of the closed leaves of the characteristic foliation. The construction and proof goes along the same lines as in the K-contact case, see [22,Section 4].
Let (M 2n+s , f, ξ α , η α , g) be compact metric f -K-contact manifold. Observe that, as ξ 1 , . . . , ξ s commute with each other, the closure in the isometry group Isom(M, g) of M of the subgroup generated by the flow of the characteristic vector fields T := exp (t 1 ξ 1 ), . . . , exp(t s ξ s ) is a connected, abelian Lie subgroup of Isom(M, g), which is also compact being M compact by hypothesis; hence T is a torus. Let Z ∈ Lie(T ) =: t be a generic element, in the following sense: in every point p ∈ M the isotropy Lie algebra t p is of dimension at most dim T −s, as the elements ξ α are never contained in it. We definet As M is compact, there are in total only finitely many distinct subspaces t p ⊂ t. We choose Z to satisfy note that this condition is void in case dim T = s.
Consider the real-valued map S : M → R, p → η α (Z)(p). Using (5.1), we have: (dS)(p) = d(i Z η α )(p) = (i Z dη α )(p) = dη α (Z p , ·). Thus the critical set C of S consists of the points p ∈ M such that Z p ∈ α R(ξ α ) p . Observe that by our choice of Z we have which is the same as the union of the closed leaves of the characteristic foliation F of M .
where k α = η α (Z)(p), α ∈ {1, . . . , s}, which vanishes along N . Then for all v, w ∈ T p M perpendicular to N we have: . Therefore the Hessian of S along N is nondegenerate in directions perpendicular to N . Proof.
The T -isotropy Lie algebra is constant along the closed submanifold N ; in fact, N is equal to a connected component of the fixed point set of T 0 p , the identity component of the isotropy group T p . It follows that t = t p ⊕ s α=1 Rξ α , and the equality Z = δ + s α=1 k α ξ α is precisely the decomposition of Z according to this decomposition of t. This implies that (2.5)) and that δ vanishes along all of N .
If ∇ v δ = 0 then, because also δ(p) = 0 and δ is a Jacobi vector field along the geodesic γ with initial velocity v, we have that δ vanishes along γ. On the other hand, by our choice of Z, in a neighborhood of N the vector field Z vanishes only in points of N . This implies that im(γ) ⊂ N , contradicting the the fact that v is perpendicular to N . To complete the proof of (i), consider a vector X at p tangent to N . As N is a closed totally geodesic submanifold of M (see [18]) and δ is a Killing vector field which, restricted to N , is tangent to N , we have g(∇ v δ, X) = −g(∇ X δ, v) = 0 for all X tangent to N , so that ∇ v δ is perpendicular to N .
To prove (ii) consider V , W local vector fields extending v, w by parallel translation along geodesics emanating from p. Using (2.5), (2.6) and the fact that Z and ξ α are Killing vector fields commuting with each other we obtain: We also compute where we used (2.6) in the second and (2.5) in the fourth equality. The last equality is true because N is T -invariant and hence v is contained in the image of f . Now, using this information, we continue the computation above, taking w = f (∇ v δ): In this computation we used that δ vanishes in p for the second equality, that ∇ v δ is perpendicular to N in the second and third equality, and the identity ∇ v Z = −kf (v) + ∇ v δ from (i) in the third equality.
Therefore it follows: The function S is a T -invariant Morse-Bott function with critical set C.

Closed leaves of the characteristic foliation
In this section we relate the ordinary and basic cohomology of a compact metric f -K-contact manifold (M 2n+s , f, ξ α , η α , g) to the union C of the closed leaves of the characteristic foliation F. This generalizes results from [22] and [15].
As usual we denote the fundamental 2-form of M 2n+s by ω. The function S considered in Section 5 is F-basic, i.e., constant along leaves of F. Because of Proposition 5.2, [17, Theorems 6.3 and 6.4] are applicable and we obtain: Theorem 6.1. We have the following equality of Poincaré polynomials: where N runs over the connected components of C, and λ N is the index of N , i.e., the rank of the negative normal bundle of N with respect to S. Proof. It suffices to show the claim for k = n. The given form is F-basic and thus defines an element in H 2k (M, F). To show that this element is nonzero we show that it maps to a nonzero real number under the above integration operator, with respect to an appropriate relatively closed s-form.
The form η 1 ∧ · · · ∧ η s is relatively closed with respect to the foliation F: we have and ω vanishes on vector fields tangent to F. Note also that M admits a natural orientation induced by the volume form ω n ∧ η 1 ∧ · · · ∧ η s . Then For s = 1, i.e., in the K-contact setting, the following theorem was known previously -the statement about the minimal number of closed leaves generalizes [22,Corollary 1], and the equivalence of the four conditions results from [15]. Theorem 6.4. The characteristic foliation of a compact metric f -K-contact manifold M 2n+s has at least n + 1 closed leaves. If it has only finitely many closed leaves, then the following conditions are equivalent: • The number of closed leaves of F is n + 1. Proof. If the number of closed leaves is finite, then it is, by Corollary 6.2, given by dim H * (M, F). But this vector space contains, by Lemma 6.3, the n + 1 nontrivial elements 1, [ω], . . . , [ω] n . This shows that there are at least n + 1 closed leaves, and the equivalence of the first and second condition.
The equivalence of the second and third condition follows from the long exact Gysin-type sequence in Proposition 4.2. The equivalence of the third and fourth condition is Theorem 4.3. Corollary 6.5. Let M 2n+1 be a real cohomology sphere, equipped with a Kcontact structure with finitely many closed Reeb orbits, and φ : M → M an automorphism of the K-contact structure. Then φ sends every closed Reeb orbit to itself.
Proof. As shown in Section 3 the mapping torus M φ of φ naturally is a metric f -K-contact manifold. Let C ⊂ M be the union of the closed Reeb orbits of M , which are exactly n + 1 by Theorem 6.4. Then the union of the closed leaves of the characteristic foliation F of M φ naturally is the mapping torus C φ , whose number of connected components is bounded from above by n+1, with equality if and only if φ sends every closed Reeb orbit to itself. But on the other hand every closed leaf of F is isolated, hence this number of connected components is by Theorem 6.4 also bounded from below by n + 1.
Example 6.6. It was shown in [15] that there is a K-contact structure with exactly four closed Reeb orbits on the 7-dimensional Stiefel manifold SO(5)/SO(3), which is a real cohomology sphere. All (iterated) mapping tori of automorphisms of this example thus satisfy the conditions in Theorem 6.4.