Metric f -Contact Manifolds Satisfying the ( κ , µ ) -Nullity Condition

: We prove that if the f -sectional curvature at any point of a ( 2 n + s ) -dimensional metric f -contact manifold satisfying the ( κ , µ ) nullity condition with n > 1 is independent of the f -section at the point, then it is constant on the manifold. Moreover, we also prove that a non-normal metric f -contact manifold satisfying the ( κ , µ ) nullity condition is of constant f -sectional curvature if and only if µ = κ + 1 and we give an explicit expression for the curvature tensor ﬁeld in such a case. Finally, we present some examples.


Introduction
Riemannian manifolds with a complementary structure adapted to the metric have been widely studied, for instance, almost complex and almost contact manifolds. Both almost complex and almost contact structures are particular cases of f -structures introduced by K. Yano [1].
For manifolds with an f -structure, D.E. Blair [4] introduced the notion of K-manifolds and their particular cases of S-manifolds and C-manifolds and proved that the space of a principal toroidal bundle over a Kaehler manifold is an S-manifold.
Moreover, S-structures are a natural generalization of Sasakian structures. However, unlike Sasakian manifolds, no S-structure can be realized on a simply connected compact manifold [7] (see also (Corollary 4.3,[8])). In [9], an example of an even dimensional principal toroidal bundle over a Kaehler manifold which does not carry any Sasakian structure is presented and an S-structure on the even dimensional manifold U(2) is constructed. Consequently and since it is well known that U(2) does not admit a Kaehler structure, there exist manifolds such that the best structure which one can hope to obtain on them is an S-structure. In this context, it seems to be necessary to generalize to the setting of metric f -manifolds the concepts and results concerning almost contact geometry.
Following this idea, B. Capelletti Montano and L. Di Terlizzi generalize the concept of contact metric manifolds such that the characteristic vector field belongs to the (κ, µ)-distribution, where κ and µ are real constants, studied in [10] and classified in [11], to metric f -contact manifolds (see [12]) by defining f -(κ, µ) manifolds. The purpose of this paper is to find conditions which characterizes f -(κ, µ) manifolds with constant f -sectional curvature. We shall prove that if the f -sectional curvature at a point p of a (2n + s)-dimensional f -(κ, µ) manifold with n > 1 is independent of the f -section al p, then it is constant on the manifold. This result is analogous to Schur's lemma in Riemannian geometry and extends a corresponding result on S-manifolds. Moreover, we shall also prove that an f -(κ, µ) manifold which is not an S-manifold is of constant f -sectional curvature if and only if µ = κ + 1 and an explicit expression for the curvature tensor field will be given.
These results generalize the corresponding ones in the case s = 1, that is, in contact geometry [13]. Finally, we shall present some examples as application of the above results.
Since f is of rank 2n, then η 1 ∧ · · · ∧ η s ∧ F n = 0 and, in particular, M is orientable. A metric f -contact manifold is said to be a metric f -K-contact manifold if the structure vector fields are Killing vector fields.
The f -structure f is said to be normal if where [ f , f ] denotes the Nijenhuis tensor of f given by for any X, Y ∈ X (M).
On a metric f -contact manifold there are defined the (1,1)-tensor fields (see [2]), where L ξ α f is the Lie derivative of f in the direction ξ α . These operators are self adjoint, traceless, anticommute with f and Moreover, the structure vector field ξ a is a Killing vector field if and only if h α = 0 (Theorem 2.6, [2]).
In the same paper [12], it is also proved that a metric f -contact manifold is an S-manifold if and only if κ = 1.
For later use, we recall that, from Lemma 2.7 of [12], in a metric f -contact manifold satisfying the (κ, µ)-nullity condition, for any X, Y, Z ∈ X (M). Moreover, in the same conditions, if Q denotes the Ricci operator and κ < 1, from Corollary 2.1 of [12]:
If for every point p ∈ M the f -sectional curvature at p is constant, then it is constant on M (so, M is called a f -(κ, µ)-space-form) and the curvature tensor of M is given by where H is the constant f -sectional curvature of M. In particular if κ < 1, then µ = κ + 1 and H = −s(2κ + 1).
Let p ∈ M and X, Y ∈ L p . Then, using basic curvature identities, Equations (4) and (7) and the fact that M a is metric f -contact manifold, we obtain: Analogously we have and: If H(p) denotes the value of the f -sectional curvature at p, then for any X, Y ∈ L p : Summing these equations and using (10), (11), (12), we get: Now we replace Y by f Y in the previous equation, obtaining where we have used (3) and (4) for the first equality and (11) and (13) for the second. Combining (14) and (15), we deduce:

Now, using this identity in
where X, Y, Z ∈ L p , we obtain, by a straightforward calculation: It is easy to check, by using the (5), that the previous equation also holds for any Z ∈ T p M and X, Y ∈ L p . Thus, for each X, Y ∈ L p : Next, let X, Y, Z ∈ L p . Then, using the above identity in we obtain: Replacing X by Y and Y by −X in (16) we have: Summing (16) and (17) and by using the Bianchi's first identity and the fact that f h is a symmetric operator and f is antisymmetric, we obtain: At this point, again one can easily check, using (5) and (4), that the above equation is also valid for any Z ∈ T p M and X, Y ∈ L p . Now, we consider any X, Y, Z ∈ T p M. We have that where X H , Y H ∈ L p and that: Consequently, using (18) and (6) in the above equation, we finally obtain (9). Next, we prove that the f -sectional curvature of M is constant. Let {e i } be a local orthonormal basis of tangent vector fields on U ⊂ M. Then, taking Y = Z = e i in (9) and summing over i we obtain the following identity for the Ricci operator at p ∈ U, where we have used (2), (1), (4), the antisymmetry of f and the fact that, since h is traceless, then, and: Comparing (19) and (8) we have that (n + 1)H(p) = s(n − 1 − 2µn − 2κ) (20) and the f -sectional curvature is constant. By (Theorem 2.3, [12]) we know that for any X ∈ L + with g(X, X) = 1. Thus, Equation (20) becomes namely (κ − µ + 1)(n − 1) = 0. Hence, since n > 1, we have that µ = κ + 1 and H = −s(2κ + 1).
To that end, firstly we state the following lemma which is proved by using (Theorem 2.2, [12]), (3) and a long straightforward computation.

Lemma 1.
Let M be an f -(κ, µ) manifold which is not an S-manifold. Let X ∈ L be a unit vector field and put X = X + + X − , where X + ∈ L + and X − ∈ L − . Then, where H(X) = K(X, f X) denotes the f -sectional curvature determined by a unit vector field X ∈ L.
Thus, we have: Proof. We only need to prove that if µ = κ + 1, then M has constant f -sectional curvature. But, from (21) we obtain H(X) = −s(2k + 1), for any unit vector field X ∈ L.

Examples
Example 1. Generalized S-space-forms. A metric f -manifold with two structure vector fields is said to be a generalized S-space form if there exist seven differentiable functions on M, F 1 , . . . , F 7 such that the curvature tensor field R of M satisfies for any X, Y, Z ∈ X (M) [17,18]. In this context, S-space-forms and C-space-forms are generalized S-space-forms. Moreover, any pseudo-umbilical, totally umbilical or totally geodesic hypersurface isometrically immersed in a generalized Sasakian-space-form [19] is also a generalized S-space-form. Then, if M is also a metric f -contact manifold, a direct expansion from (22) shows that M is a f -(κ, µ) In such a case, κ = F 1 − F 3 and µ = 0. Some examples of generalized S-space-forms satisfying these conditions can be found in [17].
Author Contributions: All the authors have contributed to the paper in the same way. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.