NEARLY SASAKIAN MANIFOLDS REVISITED

. We provide a new, self-contained and more conceptual proof of the result that an almost contact metric manifold of dimension greater than 5 is Sasakian if and only if it is nearly Sasakian.


Introduction
A Sasakian manifold M is a contact metric manifold that satisfies a normality condition, encoding the integrability of a canonical almost complex structure on the product M × R. Several equivalent characterizations of this class of manifolds, in terms of Riemannian cone, or transversal structure, or curvature, are also known. In particular one can show that an almost contact metric structure (g, φ, ξ, η) is Sasakian if and only if the covariant derivative of the endomorphism φ satisfies (1) (∇ X φ)Y − g(X, Y )ξ + η(Y )X = 0, for all vector fields X, Y ∈ Γ(T M ). A relaxation of this notion was introduced by Blair, Showers and Yano in [2], under the name of nearly Sasakian manifolds, by requiring that just the symmetric part of (1) vanishes. Later on, several important properties of nearly Sasakian manifolds were discovered by Olszak ([6]). Nearly Sasakian manifolds may be considered as an odd-dimensional analogue of nearly Kähler manifolds. In fact, the prototypical example of nearly Sasakian manifold is the 5-sphere as totally umbilical hypersurface of S 6 , endowed with the almost contact metric structure induced by the well-known nearly Kähler structure of S 6 . Nevertheless, in recent years several differences between nearly Sasakian and nearly Kähler geometry were pointed out. In particular, in [3] it was proved that the 1-form η of any nearly Sasakian manifold is necessarily a contact form, while the fundamental 2-form of a nearly Kähler manifold is never symplectic unless the manifold is Kähler. A peculiarity of nearly Sasakian five dimensional manifolds, which are not Sasakian, is that upon rescaling the metric one can define a Sasaki-Einstein structure on them. In fact one has an SU(2)-reduction of the frame bundle. Conversely, starting with a five dimensional manifold with a Sasaki-Einstein SU(2)structure it is possible to construct a one-parameter family of nearly Sasakian non-Sasakian manifolds. Thus the theory of nearly Sasakian non-Sasakian manifolds is essentially equivalent to the one of Sasaki-Einstein manifolds.
Concerning other dimensions, there have been many attempts of finding explicit examples of proper nearly Sasakian non-Sasakian manifolds until the recent result obtained in [4] showing that every nearly Sasakian structure of dimension greater than five is always Sasakian. Such result depends on the early work [3] by the first and third authors, which in turn draws many properties proved in [6]. This makes the proof to be spread over several different texts with different notation.
The aim of this note is to provide a complete and streamlined proof of the aforementioned dimensional restriction on nearly Sasakian non-Sasakian manifolds. We will also pinpoint where the positivity of the Riemannian metric is used. For this purpose we work in the more general setting of pseudo-Riemannian geometry. We will always assume that the metric is non-degenerate.
This paper was written on occasion of the conference RIEMain in Contact, held in Cagliari (Italy), 18-22 June 2018.

Preliminaries
2.1. Tensor calculus notation. In this section review the notation for the tensor calculus we use throughout the paper.
Let ∇ be a covariant derivative. It is easy to show that ∇σ = 0. If T is an arbitrary (p, q)-tensor, then ∇T can be considered as a (p, q + 1)-tensor. We define recursively the (p, q + k)-tensors ∇ k T by ∇ k+1 T := ∇(∇ k T ).
We will use the following convention regarding the arguments of ∇ k T (∇ k T )(X 1 ⊗ · · · ⊗ X q+k ) := (∇ k X1,...,X k T )(X k+1 ⊗ · · · ⊗ X q+k ). Given T 1 and T 2 of valencies (p 1 , q 1 ), (p 2 , q 2 ), respectively, and such that q 1 ≥ p 2 , we define the tensor T 1 • T 2 of type (p 1 , q 1 − p 2 + q 2 ) by Note that with our convention for ∇T , if T 1 and T 2 are tensors of valencies (p 1 , q 1 ) and (p 2 , q 2 ) respectively, then where we used the cycle notation for permutations, as we will do throughout the paper. Moverover, one has . Then (2) should be used with caution since in the term ∇T 1 •σ, we have to consider σ as an element of Σ q1 , not as an element of Σ q1+1 . Let us denote by s the inclusion Σ q1 into Σ q1+1 defined by s(σ)(i) = σ(i − 1) + 1, i ≥ 2, s (σ) (1) = 1. Then ∇(T • σ) = ∇T • s(σ). In the computations below, we will always substitute σ with s(σ) when needed, so that if in the composition chain the tensor T of type (p, q) is followed by a permutation σ then σ is always in Σ q .

2.2.
Nearly Sasakian manifolds. The definition of Sasakian manifolds was motivated by study of local properties of Kähler manifolds. Namely, Sasakian manifold is an odd dimensional Riemannian manifold (M, g) such that the metric cone (M × R + , tg + dt 2 ) is Kähler. Sasakian manifolds can also be characterized as a subclass of almost contact metric manifolds.
From the definition it follows that φξ = 0 and η • φ = 0. By [1, Theorem 6.3] the following can be used as an alternative definition of Sasakian manifolds.
By polarizing at X the condition (4) can be restated in the form As explained in the introduction, we will work in the more general setting of pseudo-Riemannian geometry. The definitions of nearly pseudo-Sasakian and pseudo-Sasakian manifolds are the same as above with only distinction that now g is a pseudo-Riemannian metric.
Since ∇ξ is skew-symmetric, we get Next we establish that the 1-form η of any nearly Sasakian manifold is contact. We use in this proposition that the metric g is positively defined, since this permits to conclude that the square of g-skew-symmetric operator has non-positive spectrum. This is not true for a general pseudo-Riemannian metric.

Curvature properties of nearly Sasakian manifolds
In this section we reestablish curvature properties of nearly Sasakian manifolds obtained by Olszak in [6]. The main consequence of these properties, used in the rest of the paper, is an explicit formula for ∇ 2 ξ in terms of ∇ξ.
We will use the following notation for curvature tensors In particular Rξ denotes the (1, 2)-tensor on M given by (Rξ For every covariant tensor T ∈ Γ(T M ⊗k ) and endomorphism φ, In the following series of propositions we show that i φ R vanishes on every nearly pseudo-Sasakian manifold. This generalizes the Olszak's result obtained in [6] for nearly Sasakian manifolds.
Proposition 3.1. Let (M, g) be a pseudo-Riemannian manifold and φ a linear endomorphism of T M . Then the tensor i φ R has the following symmetries , the result follows from the corresponding symmetries of the curvature tensor R.
The following proposition lists a well-known property of tensors with certain symmetries (see e.g. [5, page 198]). If T (X, Y, X, Y ) = 0 for any pair of vector fields X, Y then T = 0.
In the next proposition we relate the tensors i φ R and Rφ.
Proof. The result follows from and symmetries of R.
Proof. By Proposition 3.1 the tensor i φ R has the symmetries which permit to apply From the above expression it follows that (i φ R)(X, Y, X, Y ) = 0 if and only if the form Q(X, Y ) := g((∇ 2 Y,X φ)X, Y ) satisfies Q(X, Y ) = −Q(Y, X). In the remaining part of the proof we will show that Q(X, Y ) = (1/2)dη(X, Y )g(X, Y ). Then the result follows since dη is skew-symmetric and g is symmetric.
Proposition 3.6. Let (M, g) be a pseudo-Riemannian manifold and ξ a Killing vector field on M . Then ∇ 2 ξ can be determined from Rξ, namely Proof. Since ξ is Killing, the operator ∇ξ is skew-symmetric, i.e. g • (∇ξ ⊗ Id + Id ⊗ ∇ξ) = 0. Applying ∇ to this equation we get g • ( Next denote g • ξ by η. Since  (12) and (13), we get In the next proposition we collect several partial results on the curvature tensor of a nearly pseudo-Sasakian manifold.
Polarizing at X, we get R(ξ, X, ξ, Y ) = g((∇ξ) 2 X, Y ). Therefore R ξ,X ξ = (∇ξ) 2 X. Now (14) can be written in the form To compute ∇ 2 ξ, we use the expression g • ∇ 2 ξ = (g • Rξ) • (1, 2) obtained in Proposition 3.6. We get that for any X, Y , Z ∈ Γ(T M ) The above formula is equivalent to the formula for ∇ 2 ξ in the statement of the proposition since g is non-degenerate. Now let X, Y , Z be arbitrary vector fields on M . Then Since (∇ξ) 2 is self-adjoint and g is non-degenerate, we get which is equivalent to the formula in the statement.
Proof. Throughout the proof we use that ∇ξ and ∇ 2 Y ξ are skew-symmetric operators. The first fact was proved in Proposition 2.4, and the second is its consequence.

Since the trace of a nilpotent operator is always zero and
, we conclude that the both traces in (15) are zero and therefore tr(∇ξ) 2s is a constant function for all s.
In the case of nearly Sasakian manifolds Theorem 3.8 implies the existence of a tangent bundle decomposition into a direct sum of subbundles. This decomposition will be crucial in our proof of Theorem 4.6 which gives an explicit formula for ∇φ on a nearly Sasakian manifold. Recall that by Theorem 2.5 the spectrum of (∇ξ) 2 on a nearly Sasakian manifold is non-positive. Proposition 3.9. Let (M, g, φ, ξ, η) be a nearly Sasakian manifold. Suppose 0 = λ 0 > −λ 1 > · · · > −λ ℓ are the roots of the characteristic polynomial of (∇ξ) 2 . Then T M can be written as a direct sum of pair-wise orthogonal subbundles V k ⊂ T M such that, for every 0 ≤ k ≤ ℓ, the restriction of (∇ξ) 2 to V k equals −λ k · Id.
Proof. By Proposition 2.4 the operator ∇ξ is skew-symmetric, and therefore (∇ξ) 2 is symmetric. As g is positively defined this implies that (∇ξ) 2 is diagonalizable. Denote by a k the multiplicity of −λ k in the characteristic polynomial of (∇ξ) 2 . Then, by examining the diagonal form of (∇ξ) 2 , one can see that rk((∇ξ) 2 + λ k · Id) = 2n + 1 − a k and that T M can be written as a direct sum of the subbundles V k = ker((∇ξ) 2 + λ k · Id). It is a standard fact that these subbundles are mutually orthogonal and clearly the restriction of (∇ξ) 2 to V k equals −λ k · Id.

Covariant derivative of φ
In this section we derive a rather explicit formula for ∇ X φ on a nearly pseudo-Sasakian manifold. We achieve this by computing separately ∇ X φ on subspaces ξ , Im(∇ ξ φ), and Im(∇ ξ φ) ⊥ ∩ξ ⊥ . Then, we will use the formula to prove Theorem 4.9.
Given two tensor fields T 1 and T 2 on a manifold M such that both products T 1 •T 2 and T 2 • T 1 make sense, we define commutator and anticommutator of T 1 and T 2 by T 1 , aim of the next three propositions is to find (∇ X φ)Y on a nearly pseudo-Sasakian manifold in the case Y is in the image of ∇ ξ φ. For this we compute (∇φ)(∇ ξ φ).
Proof. By Proposition 2.4, we know that ∇ ξ φ = φ(φ + ∇ξ). Notice that for any three tensors A, B, and C, such that all pair-wise compositions are defined, we have Thus to find the commutator of ∇φ with ∇ ξ φ, we only have to compute the anticommutators of ∇φ with φ and ∇ξ.
Proposition 4.4. Let (M, g, φ, ξ, η) be a nearly pseudo-Sasakian manifold. Then for any Y in the image of ∇ ξ φ, the following equation holds Proof. Let Z be such that (∇ ξ φ)Z = Y . Since (∇ ξ φ)ξ = 0 we can assume that η(Z) = 0 by replacing Z with Z − η(Z)ξ if necessary. By Proposition 4.2, we get Next, by Proposition 4.3, we have This finishes the proof.
In the next proposition we use that g is positively defined to conclude that (∇ ξ φ) 2 Y = 0 implies (∇ ξ φ)Y = 0. This can be false for a general nearly pseudo-Sasakian manifold.
Remark 4.7. It follows from (29) that a nearly Sasakian manifold is Sasakian if and only if ∇ ξ φ = 0. In fact, if ∇ ξ φ = 0, then (29) implies which is the defining condition of Sasakian structures. In the opposite direction, if M is a Sasakian manifold, then computing ∇ ξ φ by (31) we get zero.