Canonical submersions in nearly Kähler geometry

Leander Stecker

doi:10.1515/coma-2025-0016

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Canonical submersions in nearly Kähler geometry

Leander Stecker

Published/Copyright: August 26, 2025

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From the journal Complex Manifolds Volume 12 Issue 1

Abstract

We explore submersions introduced by reducible holonomy representations of connections with parallel skew torsion. A submersion theorem extending previous, less general, results is given. As our main application, we show that parallel 3- ( α , δ ) -Sasaki manifolds admit one-dimensional submersions onto nearly Kähler orbifolds. As a secondary application, we reprove that a given class of nearly Kähler manifolds submerges onto quaternionic Kähler manifolds. This new proof gives a direct expression for the quaternionic structure on the base.

Keywords: skew-torsion connection; Riemannian submersions; 3-(α, δ)-Sasaki manifolds; nearly Kähler geometry; quaternionic Kähler manifolds

MSC 2010: 53B05; 53C15; 53C25; 53C29

1 Introduction

The investigation of holonomy groups has played a key role in Riemannian geometry over the last century. Thanks to de Rham splitting, one can reduce the investigation to irreducible holonomy representations. It therefore is possible to classify them in the shape of Berger’s list of special holonomies. While these geometries are interesting and widely studied, they are narrow and many, in particular odd-dimensional, geometries fail to be represented. An early generalization due to Gray were weak holonomy groups, extending to the classes of nearly Kähler, nearly parallel G 2 , and nearly parallel Spin ( 9 ) manifolds [10,14]. We are foremost interested in the broader class of geometries that admit a metric connection with skew symmetric torsion, or geometries with skew torsion for short. These include the aforementioned classes of weak holonomies and are also of widespread interest in type II string theory (compare [11–13]).

Unlike with the Levi-Civita connection, for connections with skew torsion, there exists no general de Rham splitting so reducible representations are very much to be considered. In these cases, the reducibility tells us a great deal about the geometry. In fact, under the assumption of parallel torsion, Cleyton et al. [7] showed that they always admit a locally defined Riemannian submersion. We refine their theorem giving us control over the submersion motivated by the case of 3- ( α , δ ) -Sasaki manifolds in [3]. We will call them canonical submersions.

3- ( α , δ ) -Sasaki manifolds were defined by the condition d η i = 2 α Φ i + 2 ( α − δ ) ξ j ∧ ξ k in [2] as a generalization of 3-Sasaki manifolds when α = δ = 1 . Pivotally, they admit a canonical connection ∇ with skew torsion. In [3], it was shown that ∇ gives rise to a canonical submersion in aforementioned sense, whose base admits a quaternionic Kähler, resp. hyperkähler, structure. This divides them into positive, negative, and degenerate 3- ( α , δ ) -Sasaki structures, depending on the scalar curvature of the quaternionic Kähler base. In the positive realm, there are various interesting cases of parameters. Apart from aforementioned 3-Sasakian if α = δ = 1 , there is the secondary Einstein metric or so-called squashed 3-Sasaki Einstein metric if δ = ( 2 n + 3 ) α , [2]. We are particularly interested in the parallel case δ = 2 α . Here, ∇ parallelizes all Reeb vector fields. We will show that this yields apart from the submersion discussed in [7] and [3], another admissible submersion of codimension 1 onto nearly Kähler spaces. This is the first instance investigated of canonical submersions whose base is a geometry with non-vanishing torsion showing that this possibility is not an inconvenience, but a feature of the theory.

Nearly Kähler manifolds are almost Hermitian, non-Kähler manifolds, defined by the condition ( ∇ X J ) X = 0 . They garnered particular attention in dimension 6. In this dimension, their cone has holonomy G 2 , in [6], they admit a real Killing spinor and are Einstein. Though this is not true for dimensions greater than 6, they are connected to many geometries in any dimension. For instance, they are known to exist on the twistor spaces of quaternionic Kähler manifolds [8,9]. It therefore should not come as a surprise they can be obtained as a quotient of the Konishi space. However, our proof shows that the nearly Kähler structure and its canonical Hermitian connection are tightly locked with the parallel 3- ( α , δ ) -Sasaki structure and its canonical connection. A second application closes the circle showing how a class of nearly Kähler manifolds, essentially those obtained by the aforementioned construction, admit a further canonical submersion onto quaternionic Kähler spaces. This result was shown previously in the decomposition of nearly Kähler spaces of dim ≥ 6 by Nagy in [17] and locally in [4]. However, we include it since the proof fits neatly into our established framework of canonical submersions.

2 Canonical submersions

On a Riemannian manifold ( M , g ) , a metric connection ∇ on the tangent bundle is uniquely characterized by its torsion tensor

T ( X , Y , Z ) ≔ g ( ∇ X Y − ∇ Y X − [ X , Y ] , Z ) .

By definition, the torsion is skew-symmetric in the first two entries. If, however, it is totally skew-symmetric, we say that ∇ has skew torsion. In this case, the connection is given by

g ( ∇ X Y , Z ) = g ( ∇ X g Y , Z ) + 1 2 T ( X , Y , Z ) ,

where we denote by ∇ g the Levi–Civita connection of ( M , g ) . We require additionally that our connection ∇ has parallel torsion, i.e., ∇ T = 0 . This yields many simplifications, among them, such connections satisfy the following Bianchi identity:

(1) S X , Y , Z R ∇ ( X , Y , Z , V ) = σ T ( X , Y , Z , V ) ≔ S X , Y , Z g ( T ( X , Y ) , T ( Z , V ) ) ,

where the shorthand notation S X , Y , Z denotes the sum over all cyclic permutations of X , Y , and Z .

With those preliminaries, we pose our first main theorem. Given a manifold admitting a connection with parallel skew torsion and reducible holonomy, then the Canonical Submersion Theorem answers the question how such a manifold can be simplified. Earlier, more restrictive versions of this theorem were presented in [3,7,18]. However, since the assumptions are simplified, we include the proof.

Theorem 2.1

Suppose ∇ is a metric connection with parallel skew torsion T on ( M , g ) and T M = ℋ ⊕ V splits orthogonally as representation of the reduced holonomy group Hol 0 ( ∇ ) . Assume further that

(2) T ∈ Λ 3 ℋ ⊕ Λ 2 ℋ ∧ V ⊕ Λ 3 V ⊊ Λ 3 T M ,

i.e., the Λ 2 V ∧ ℋ -part of T vanishes.

Then, there exists a locally defined Riemannian submersion π : ( M , g ) → ( N , g N ) with totally geodesic fibers tangent to V , the purely horizontal part of the torsion T ℋ is projectable, π * T ˇ ≔ T ℋ , and ∇ T ˇ = ∇ g N + 1 2 T ˇ is a connection with parallel skew torsion on N satisfying

(3) ∇ X T ˇ Y = π * ( ∇ X ¯ Y ¯ ) ,

where X ¯ and Y ¯ denote the horizontal lifts of vector fields X , Y ∈ Γ T N .

Proof

We note that by (2) and the invariance of V under ∇ for any vertical vector fields V , W ∈ V , we have

∇ V g W = ∇ V W − 1 2 T ( V , W ) ∈ V .

Therefore, the distribution V is integrable and a curve on the integral submanifold tangent to V is a geodesic if and only if it is a geodesic in M . The integral submanifolds give rise to a foliation and hence to a submersion π from a small neighborhood U ⊂ M to a local transverse section S . We show that the metric restricted to ℋ × ℋ is constant along vertical vector fields and therefore projectable. For X , Y ∈ Γ ℋ , we have

( L V g ) ( X , Y ) = V ( g ( X , Y ) ) − g ( [ V , X ] , Y ) − g ( X , [ V , Y ] ) = g ( ∇ X g V , Y ) + g ( X , ∇ Y g V ) = g ( ∇ X V , Y ) + g ( X , ∇ Y V ) = 0 ,

since V is preserved by ∇ . This proves that π is a Riemannian submersion.

To prove the second assertion, we denote T ℋ = pr Λ 3 ℋ T . If we show that T ℋ is constant along the fibers, it projects to a well-defined 3-form T ˇ . Let V ∈ V . Then,

(4) S X , Y , Z T ℋ ( X , Y , T ( V , Z ) ) = 0 ,

whenever either X , Y , Z ∈ V . Indeed, by (2), we have that T ( V , Z ) ∈ V if Z ∈ V and, hence, T ( V , Z ) _ ∣ T ℋ = 0 . Now, consider X , Y , Z ∈ ℋ . Since ∇ preserves V and ℋ the curvature R ∇ ( X , Y , Z , V ) = 0 and the Bianchi identity for connections with parallel skew torsion, (1) implies

0 = − S X , Y , Z R ∇ ( X , Y , Z , V ) = − σ T ( X , Y , Z , V ) = − S X , Y , Z g ( T ( X , Y ) , T ( Z , V ) ) = S X , Y , Z T ( X , Y , T ( V , Z ) ) = S X , Y , Z T ℋ ( X , Y , T ( V , Z ) ) ,

where the last step used again that T ( V , Z ) ∈ ℋ . Thus, (4) holds for any X , Y , Z ∈ T M .

As the subspaces V and ℋ are preserved by ∇ so are the components of tensors on T M . In particular, ∇ T = 0 implies ∇ T ℋ = 0 . Use (4) and ∇ Z V ∈ V to obtain

( L V T ℋ ) ( X , Y , Z ) = V ( T ℋ ( X , Y , Z ) ) − S X , Y , Z T ℋ ( X , Y , L V Z ) = ( ∇ V T ℋ ) ( X , Y , Z ) + S X , Y , Z T ℋ ( X , Y , ∇ V Z ) − S X , Y , Z T ℋ ( X , Y , ∇ V Z − ∇ Z V − T ( V , Z ) ) = S X , Y , Z T ℋ ( X , Y , T ( V , Z ) ) = 0 .

Equation (3) follows directly from ∇ X g N Y = π * ( ∇ X ¯ g Y ¯ ) for Riemannian submersions. Finally, we conclude that T ˇ is ∇ T ˇ -parallel:

□ ( ∇ A T ˇ T ˇ ) ( X , Y , Z ) = A ( T ˇ ( X , Y , Z ) ) − S X , Y , Z T ˇ ( X , Y , ∇ A T ˇ Z ) = A ( π ∘ T ℋ ( X ¯ , Y ¯ , Z ¯ ) ) − S X , Y , Z T ˇ ( X , Y , π * ( ∇ A ¯ Z ¯ ) ) = A ¯ ( T ℋ ( X ¯ , Y ¯ , Z ¯ ) ) − S X , Y , Z T ℋ ( X ¯ , Y ¯ , ∇ A ¯ T Z ¯ ) = ( ∇ A ¯ T ℋ ) ( X ¯ , Y ¯ , Z ¯ ) = 0 .

Observe that compared to earlier versions, this yields the following well-known generalization of de Rham splitting as an immediate consequence.

Corollary 2.2

Let ∇ be a connection with parallel skew torsion T and T M = V 1 ⊕ V 2 a holonomy-invariant decomposition. Then, locally

( M , g , ∇ ) = ( M 1 , g 1 , ∇ T 1 ) × ( M 2 , g 2 , ∇ T 2 ) ,

if and only if the torsion is decomposable, i.e., T = T 1 + T 2 with T i ∈ Λ 3 V i .

Proof

Here, (2) is satisfied for V = V 1 , ℋ = V 2 , and vice versa. Hence, both distributions are integrable, and we obtain locally defined Riemannian projection maps to their respective integral submanifolds. The converse is clear.□

Remark 2.3

A splitting T M = V ⊕ ℋ satisfying (2) is not necessarily unique. In [7], it is shown that, if there exists any Hol 0 ( ∇ ) -invariant splitting, there exists one with maximal V satisfying (2). This was recently coined as the canonical splitting in [16] compared to other admissible splittings satisfying (2). Our main application in Theorem 4.1 reflects that distinction. It is crucially an admissible, but never canonical, splitting.

We now show that a further splitting of the holonomy representation can manifest a splitting on the base.

Proposition 2.4

Suppose the holonomy representation of ∇ splits into orthogonal submodules T M = V ⊕ ℋ 1 ⊕ ℋ 2 such that V and ℋ = ℋ 1 ⊕ ℋ 2 satisfy condition (2) of Theorem 2.1. Let π : M → N denote the canonical submersion, and suppose further that ℋ 1 and ℋ 2 are projectable. Then, the representation T N of the holonomy Hol 0 ( ∇ T ˇ ) is reducible into modules π * ℋ 1 ⊕ π * ℋ 2 .

Proof

By the Ambrose-Singer theorem, the holonomy algebra at x ∈ N is generated by elements of the form

( P γ ∇ T ˇ ) − 1 ∘ ℛ ∇ T ˇ ( X ∧ Y ) ∘ P γ ∇ T ˇ ,

where P γ ∇ T ˇ denotes the parallel transport along a piecewise smooth curve γ from x to p and ℛ ∇ T ˇ ( X ∧ Y ) ∈ Λ 2 T p N ⊂ End ( T p N ) is the evaluation of the curvature operator at X , Y ∈ T p N . For any such curve, let γ ¯ be the horizontal lift starting at a point x 0 ∈ π − 1 ( x ) . Then, let X ( t ) be the unique parallel vector field along γ . By (3),

π * ( ∇ γ ¯ ˙ X ( t ) ¯ ) = ∇ γ ˙ T ˇ X ( t ) = 0

and hence, ∇ γ ¯ ˙ X ( t ) ¯ ∈ V . However, ∇ γ ¯ ˙ X ( t ) ¯ ∈ ℋ since X ( t ) ¯ ∈ Γ ℋ and ℋ is invariant under the holonomy of ∇ . Therefore, ∇ γ ¯ ˙ X ( t ) ¯ = 0 and

P γ ¯ ∇ X ¯ = P γ ∇ T ˇ X ¯ .

In particular, parallel transport with respect to ∇ T ˇ preserves π * ℋ 1 and π * ℋ 2 . Now, let X , Y ∈ T p N , then

R ∇ T ˇ ( X ∧ Y ) Z = ∇ X T ˇ ∇ Y T ˇ Z − ∇ Y T ˇ ∇ X T ˇ Z − ∇ [ X , Y ] T ˇ Z = π * ( ∇ X ¯ ∇ Y ¯ Z ¯ − ∇ Y ¯ ∇ X ¯ Z ¯ − ∇ [ X , Y ] ¯ Z ¯ )

preserves the modules π * ℋ 1 and π * ℋ 2 as well.□

The following computational lemma has already been used in [3]. However, it holds for any canonical submersion.

Lemma 2.5

Let ∇ be a connection as in Theorem 2.1. Then,

g ( ∇ X Y , Z ) = T ( X , Y , Z ) ,

for any vertical vector X ∈ V , horizontal vector Z ∈ ℋ , and basic vector field Y ∈ Γ ℋ . In particular, the expression is tensorial.

Proof

Recall that a vector field Y is basic if it is horizontal and π -related to a vector field on N . In particular, we have π * [ X , Y ] = [ π * X , π * Y ] = 0 for any vector field X ∈ Γ V . Note also that ∇ Y X ∈ V as the decomposition is hol ( ∇ ) -invariant. Therefore,

□ g ( ∇ X Y , Z ) = g ( ∇ Y X , Z ) + g ( [ X , Y ] , Z ) + T ( X , Y , Z ) = T ( X , Y , Z ) .

3 Parallel 3- ( α , δ ) -Sasaki and nearly Kähler manifolds

We like to introduce the structures involved in the application of canonical submersions we consider in this article, in particular, 3- ( α , δ ) -Sasakian and nearly Kähler manifolds. An almost contact metric manifold ( M 2 n + 1 , g , ξ , η , φ ) is given by an odd-dimensional Riemannian manifold, a unit length vector field ξ ∈ Γ T M called Reeb vector field, its metric dual 1-form η , and an almost Hermitian structure φ ∈ Γ End ( T M ) on ker η satisfying

φ ξ = 0 , η ∘ φ = 0 , φ 2 = − id + ξ ⊗ η , g ( φ X , φ Y ) = g ( X , Y ) − η ( X ) η ( Y ) .

These are considered the odd-dimensional analog of almost Hermitian manifolds. As for the latter, we define the fundamental 2-form Φ ( X , Y ) = g ( X , φ Y ) for a given almost contact metric manifold.

A tuple ( M , g , ξ i , φ i , η i ) i = 1,2,3 of three almost contact metric structures on the same underlying Riemannian manifold is called almost 3-contact metric manifold if they additionally satisfy the compatibility conditions

φ i ξ j = ξ k , η i ∘ φ j = η k , φ i φ j = φ k + ξ i ⊗ η j .

These properties guarantee that the endomorphisms act as imaginary quaternions on the horizontal distribution ℋ ≔ ⋂ ker η i . Accordingly, the complementary distribution V ≔ ℋ ⊥ = span { ξ 1 , ξ 2 , ξ 3 } is called vertical.

Definition 3.1

[2] An almost 3-contact metric manifold is called 3- ( α , δ ) -Sasaki manifold for real constants δ and α ≠ 0 if

(5) d η i = 2 α Φ i + 2 ( α − δ ) η j ∧ η k ,

for every even permutation ( i j k ) of ( 123 ) .

Most prominently, if α = δ = 1 , the manifold is 3-Sasakian. A second class often considered is the second Einstein metric with parameters δ = ( 2 n + 3 ) α , where dim M = 4 n + 3 . Our work highlights the subclass of parallel 3- ( α , δ ) -Sasaki manifolds, i.e., if δ = 2 α . Their most prominent feature is highlighted with regard to the canonical connection as defined via the following theorem.

Theorem 3.2

[2] A 3- ( α , δ ) -Sasaki manifold admits a unique metric connection ∇ with skew torsion such that

(6) ∇ X φ i = β ( η k ( X ) φ j − η j ( X ) φ k ) ,

for every even permutation ( i j k ) of (123) and with β = 2 ( δ − 2 α ) .

The connection ∇ preserves the splitting T M = V ⊕ ℋ , and its torsion is given by

(7) T = 2 α ∑ i = 1 3 η i ∧ Φ i − 2 ( α − δ ) η 123 = 2 α ∑ i = 1 3 η i ∧ Φ i ℋ + 2 ( δ − 4 α ) η 123 ,

where Φ i ℋ ≔ Φ i ∣ ℋ . In particular, the torsion is parallel ∇ T = 0 .

Hence, for parallel 3- ( α , δ ) -Sasaki manifolds, the canonical connection parallelizes all structure tensors. On homogeneous parallel 3- ( α , δ ) -Sasaki manifolds, it was shown that the metric is naturally reductive with respect to the standard presentation as a quotient of the automorphism group (see [3]). In that case, the canonical and Ambrose-Singer connections coincide.

The canonical connection also gave rise for an initial instance of canonical submersions over quaternionic Kähler spaces in [3]. Recall that quaternionic Kähler manifolds ( N ˇ 4 n , g N ˇ , Q ) , n ≥ 2 are equipped with a three-dimensional subbundle Q ⊂ End ( T N ˇ ) locally generated by a triple of almost Hermitian structures that is invariant under Hol ( ∇ g N ˇ ) . Equivalently, they are the class of Riemannian manifolds with exceptional holonomy given by a subgroup of Sp ( 1 ) Sp ( n ) . For dim N ˇ = 4 , we require N ˇ to be Einstein and anti-self-dual.

Theorem 3.3

[3] Let ( M , g , ξ i , φ i , η i ) i = 1,2,3 be a 3- ( α , δ ) -Sasaki manifold. Then, there exists a locally defined Riemannian submersion π : M → N ˇ with totally geodesic fibers tangent to V such that ∇ X g N ˇ Y = π * ( ∇ X ¯ Y ¯ ) . The base space N ˇ admits a quaternionic Kähler structure locally generated by the three almost Hermitian structures

I i = π * ∘ φ i ∘ s * ,

for any locally defined section s : N ˇ → M .

We should remark that in the following, we will fix ℋ and V to be the horizontal and vertical distributions for a 3- ( α , δ ) -Sasaki manifold, while the horizontal/vertical subspaces in further applications of Theorem 2.1 will be denoted appropriately to the situation.

For later use, we recall the following Lie derivatives along vertical vectors.

Proposition 3.4

[2] For any 3- ( α , δ ) -Sasaki manifold and any even permutation ( i j k ) of ( 123 ) , we have the following identities:

(8) ℒ ξ i φ i = 0 , ℒ ξ i φ j = − ℒ ξ j φ i = 2 δ φ k ,

(9) ℒ ξ i ξ i = 0 , ℒ ξ i ξ j = − ℒ ξ j ξ i = 2 δ ξ k .

With that, let us shift attention to nearly Kähler manifolds.

Definition 3.5

A nearly Kähler manifold ( N 2 m , g N , J ) is an almost Hermitian manifold such that ( ∇ X g N J ) X = 0 for all X ∈ T N .

As initially observed in [15], a nearly Kähler manifold admits a particularly nice connection ∇ c . We will call ∇ c the characteristic connection of a nearly Kähler manifold as it is the unique connection with skew torsion preserving the U ( m ) -structure (compare [1]). In the literature, it is often also called Bismut or canonical connection. In fact, their usual definitions agree on nearly Kähler manifolds as it is the unique Hermitian connection and has torsion

T c ( X , Y , Z ) = g N ( ( ∇ X g N J ) J Y , Z ) .

Furthermore, T c is parallel with respect to ∇ c .

4 Nearly Kähler orbifolds from parallel 3- ( α , δ ) -Sasaki manifolds

Let ( M , g , ξ i , φ i , η i ) i = 1,2,3 be a 3- ( α , δ ) -Sasaki manifold and ∇ its canonical connection with parallel skew torsion T . Choose any almost contact structure φ in the associated sphere Σ = { a 1 φ 1 + a 2 φ 2 + a 3 φ 3 ∣ ∣ a ∣ = 1 } , and denote its corresponding Reeb vector field ξ and contact form η . If β = 0 , or equivalently, δ = 2 α , we have ∇ φ = 0 and also ∇ ξ = 0 and ∇ η = 0 . Hence, the decomposition T M = ⟨ ξ ⟩ ⊕ ker η = ⟨ ξ ⟩ ⊕ ⟨ ξ ⟩ ⊥ is holonomy-invariant. Note that (2) is trivially satisfied for any one-dimensional vertical distribution. In the following, we apply Theorem 2.1 to this setup. We fix the notation V and ℋ for the vertical and horizontal spaces of the initial parallel 3- ( α , δ ) -manifold and denote the vertical and horizontal spaces with respect to this newly obtained submersion by ⟨ ξ ⟩ and ⟨ ξ ⟩ ⊥ , respectively.

Theorem 4.1

Let ( M , g , ξ i , φ i , η i ) i = 1,2,3 be a parallel 3- ( α , δ ) -Sasaki manifold and fix an almost contact metric structure ( ξ , φ , η ) inside the associated sphere.

Then, there exists a locally defined Riemannian submersion π : ( M , g ) → ( N , g N ) along the orbits of ξ .
Set φ ˜ ≔ φ ∣ ℋ − φ ∣ V and J = π * ∘ φ ˜ ∘ s * with an arbitrary local section s : N → M of π . Then, ( N , g N , J ) is nearly Kähler.
The characteristic connection on ( N , g N , J ) agrees with the connection obtained from the canonical connection on M. In particular, T ˇ ( X , Y , Z ) = g N ( ( ∇ X g N J ) J Y , Z ) .

Proof

We have already seen that the assumptions in Theorem 2.1 are satisfied. We may in the following assume ξ = ξ 1 . Theorem 2.1, then implies that there is a locally defined Riemannian submersion π : ( M , g ) → ( N , g N ) along the orbits of ξ 1 and horizontal space ⟨ ξ 1 ⟩ ⊥ = ⟨ ξ 2 , ξ 3 ⟩ ⊕ ℋ . Furthermore, there is a connection ∇ T ˇ = ∇ g N + 1 2 T ˇ on T N with parallel skew torsion given by

(10) π * T ˇ = T ⟨ ξ ⟩ ⊥ = 2 α ( η 2 ∧ Φ 2 ℋ + η 3 ∧ Φ 3 ℋ ) ,

where we have used (7). This connection satisfies

∇ X T ˇ Y = π * ( ∇ X ¯ Y ¯ ) .

Now, consider J = π * ∘ φ ˜ ∘ s * . Due to (8), we have ℒ ξ φ = 0 and since ℒ ξ preserves ℋ and V , also ℒ ξ φ ˜ = 0 . In particular, J is independent of the choice of s . The compatibility with g N follows immediately as π * : ⟨ ξ ⟩ ⊥ → T N and pr ⟨ ξ ⟩ ⊥ ∘ s * : T N → ⟨ ξ ⟩ ⊥ are isometric. We check that J 2 = − id . Since s is a section of π , we have s * ∘ π * = id on the image of s , and thus,

J 2 = π * ∘ φ ˜ ∘ s * ∘ π * ∘ φ ˜ ∘ s * = π * ∘ φ ˜ 2 ∘ s * = π * ∘ ( − id + η ⊗ ξ ) ∘ s * = − id ,

where we have used that all involved endomorphisms preserve the orthogonal splitting ℋ ⊕ ⟨ ξ 2 , ξ 3 ⟩ .

We check that J is parallel with respect to ∇ T ˇ . Remark that the horizontal lift of any vector field on T N is basic and π * ( φ ˜ 1 ( s * Y ) ) ¯ = ( φ ˜ 1 ( s * Y ) ) ℋ ⊕ ⟨ ξ 2 , ξ 3 ⟩ wherever the right side is defined, i.e., on the image s ( N ) . Set X ˆ ≔ X ¯ − s * X the vertical part of − s * X . Then,

( ∇ X T ˇ J ) Y = ∇ X T ˇ ( J Y ) − J ( ∇ X T ˇ Y ) = π * ( ∇ X ¯ J Y ¯ ) − J ( π * ( ∇ X ¯ Y ¯ ) ) = π * ( ∇ X ¯ ( π * ( φ ˜ 1 ( s * Y ) ) ¯ ) − φ ˜ 1 ( s * ( π * ( ∇ X ¯ Y ¯ ) ) ) ) = π * ( ∇ X ˆ ( π * ( φ ˜ 1 ( s * Y ) ) ¯ ) + ∇ s * X ( π * ( φ ˜ 1 ( s * Y ) ) ¯ ) − φ ˜ 1 ( ∇ X ˆ Y ¯ ) − φ ˜ 1 ( ∇ s * X Y ¯ ) ) .

Remark that the horizontal lift of any vector field on T N is basic so we may employ Lemma 2.5. In our case,

g ( ∇ ξ 1 H , Z ) = T ( ξ 1 , H , Z ) = 2 α ∑ i = 1 3 η i ∧ Φ i ℋ − 2 η 123 ( ξ 1 , H , Z ) = 2 α ( Φ 1 ℋ ( H , Z ) − 2 η 23 ( H , Z ) ) ,

where H is either π * ( φ ˜ 1 ( s * Y ) ) ¯ or Y ¯ . Note that we apply π * in the end so it suffices to assume Z ∈ ⟨ ξ 1 ⟩ ⊥ in the following:

g ( ∇ X ˆ ( π * ( φ ˜ 1 ( s * Y ) ) ¯ , Z ) ) = 2 α η 1 ( X ˆ ) ( Φ 1 ℋ ( π * ( φ ˜ 1 ( s * Y ) ) ¯ , Z ) − 2 η 23 ( π * ( φ ˜ 1 ( s * Y ) ) ¯ , Z ) ) = 2 α η 1 ( X ˆ ) ( Φ 1 ℋ ( φ ˜ 1 ( s * Y ) , Z ) − 2 η 23 ( φ ˜ 1 ( s * Y ) , Z ) ) = 2 α η 1 ( X ˆ ) ( Φ 1 ℋ ( φ 1 ( s * Y ) , Z ) + 2 η 23 ( φ 1 ( s * Y ) , Z ) ) = 2 α η 1 ( X ˆ ) ( g ( ( s * Y ) ℋ , Z ) − 2 ( η 3 ( s * Y ) η 3 ( Z ) + η 2 ( s * Y ) η 2 ( Z ) ) ) = 2 α η 1 ( X ˆ ) ( g ( ( s * Y ) ℋ , Z ) − 2 g ( ( s * Y ) ⟨ ξ 2 , ξ 3 ⟩ , Z ) ) , g ( φ ˜ 1 ( ∇ X ˆ Y ¯ ) , Z ) = − 2 α η 1 ( X ˆ ) ( Φ 1 ℋ ( Y ¯ , φ ˜ 1 Z ) − 2 η 23 ( Y ¯ , φ ˜ 1 Z ) ) = − 2 α η 1 ( X ˆ ) ( Φ 1 ℋ ( Y ¯ , φ 1 Z ) + 2 η 23 ( Y ¯ , φ 1 Z ) ) = 2 α η 1 ( X ˆ ) ( g ( Y ¯ , Z ) − 2 ( η 2 ( Y ¯ ) η 2 ( Z ) + η 3 ( Y ¯ ) η 3 ( Z ) ) ) = 2 α η 1 ( X ˆ ) ( g ( ( Y ¯ ) ℋ , Z ) − 2 g ( ( Y ¯ ) ⟨ ξ 2 , ξ 3 ⟩ , Z ) ) .

Since ( s * Y ) ℋ ⊕ ⟨ ξ 2 , ξ 3 ⟩ = Y ¯ , both terms cancel. Furthermore, both φ ˜ 1 and ∇ preserve the splitting T M = R ξ 1 ⊕ ⟨ ξ 2 , ξ 3 ⟩ ⊕ ℋ ; thus, ∇ s * X s * Y − ∇ s * X Y ¯ is vertical and

(11) ( ∇ X T ˇ J ) Y = π * ( ∇ s * X ( φ ˜ 1 ( s * Y ) ) − φ ˜ 1 ( ∇ s * X s * Y ) ) = π * ( ( ∇ s * X φ ˜ 1 ) s * Y ) = 0 .

In order to control the covariant derivative of J with respect to the Levi-Civita connection, we need to compute

g ( ( T ˇ X ⋅ J ) Y , Z ) = g ( T ˇ X ⋅ ( J Y ) , Z ) − g ( J ( T ˇ X ⋅ Y ) , Z ) = T ˇ ( X , J Y , Z ) + T ˇ ( X , Y , J Z ) .

This is linear so we may compute it for any combination of X , Y , Z in either π * ℋ or π * V individually. Note that J preserves this splitting and π * T ˇ ∈ V ∧ Λ 2 ℋ by (10). Thus, we only need to check for these combinations of vectors. Furthermore, note that g ( ( T ˇ X ⋅ J ) Y , Z ) is skew-symmetric in Y and Z . Two cases remain. For X ∈ π * V and Y , Z ∈ π * ℋ , we have

(12) g ( ( T ˇ X ⋅ J ) Y , Z ) = T ˇ ( X , J Y , Z ) + T ˇ ( X , Y , J Z ) = 2 α ∑ i = 2,3 η i ( X ¯ ) ( Φ i ( φ 1 s * Y , Z ¯ ) + Φ i ( Y ¯ , φ 1 s * Z ) ) = 2 α ∑ i = 2,3 η i ( X ¯ ) ( g ( Y ¯ , φ i φ 1 Z ¯ ) − g ( Y ¯ , φ 1 φ i Z ¯ ) ) = − 4 α ( η 2 ( X ¯ ) Φ 3 ( Y ¯ , Z ¯ ) − η 3 ( X ¯ ) Φ 2 ( Y ¯ , Z ¯ ) ) .

For X , Z ∈ π * ℋ and Y ∈ π * V ,

(13) g ( ( T ˇ X ⋅ J ) Y , Z ) = T ˇ ( X , J Y , Z ) + T ˇ ( X , Y , J Z ) = − 2 α ∑ i = 2,3 ( η i ( φ ˜ 1 s * Y ) Φ i ( X ¯ , Z ¯ ) + η i ( Y ¯ ) Φ i ( X ¯ , φ 1 s * Z ) ) = − 2 α ( η 3 ( Y ¯ ) Φ 2 ( X ¯ , Z ¯ ) − η 2 ( Y ¯ ) Φ 3 ( X ¯ , Z ¯ ) − η 2 ( Y ¯ ) Φ 3 ( X ¯ , Z ¯ ) + η 3 ( Y ¯ ) Φ 2 ( X ¯ , Z ¯ ) ) = 4 α ( η 2 ( Y ¯ ) Φ 3 ( X ¯ , Z ¯ ) − η 3 ( Y ¯ ) Φ 2 ( X ¯ , Z ¯ ) ) .

This implies that ( ∇ X g N J ) X = ( ∇ X T ˇ J ) X − 1 2 ( T ˇ X ⋅ J ) X = 0 . Indeed, (13) proves that the π * ℋ × π * ℋ -part is skew, and the sign difference between (12) and (13) shows that we are skew for mixed terms as well. Therefore, ( N , g N , J ) is nearly Kähler.□

Corollary 4.2

The locally defined Riemannian submersion π gives rise to a globally defined submersion π : M → N , where ( N , g N , J ) is a nearly Kähler orbifold.

Proof

We need to prove that the R -action generated by ξ is locally free. Now, ξ generates a one-dimensional subgroup of the group SU ( 2 ) generated by V . Therefore, the orbits of ξ are compact S 1 , in particular, the action is locally free.□

Coming from parallel 3- ( α , δ ) -Sasaki manifolds, these nearly Kähler spaces are rather special, inheriting additional properties.

Proposition 4.3

The nearly Kähler spaces obtained through Theorem 4.1 have reducible characteristic holonomy.

Proof

As in the proof of Theorem 4.1, we may assume that ξ = ξ 1 . We show that the holonomy representation T M = ⟨ ξ 1 ⟩ ⊕ ⟨ ξ 2 , ξ 3 ⟩ ⊕ ℋ satisfies the conditions of Proposition 2.4. Since ∇ preserves each Reeb vector field individually, the aforementioned decomposition is Hol 0 ( ∇ ) -invariant. By (9), the distribution ⟨ ξ 2 , ξ 3 ⟩ is invariant under ξ 1 and, thus, projectable. As ξ 1 is Killing, the same is true for ℋ .□

Remark 4.4

This shows that projectability is essential in Proposition 2.4 as both ξ 2 and ξ 3 are parallel with respect to ∇ but their projections individually are not.

Complete strictly nearly Kähler 6-folds with reducible characteristic holonomy were investigated by Belgun and Moroianu [5]. They show that the only such manifolds are the twistor spaces C P 3 and F ( 1,2 ) with their standard nearly Kähler structures. More generally, we see that ( N , g N , J ) is of special algebraic torsion in the notation of [17]. That is, T ˇ ( X , Y ) = 0 for X , Y ∈ π * ⟨ ξ 2 , ξ 3 ⟩ and T ˇ ( X , Y ) ⊂ π * ⟨ ξ 2 , ξ 3 ⟩ for any X , Y ∈ ℋ . Indeed, from (10), the projections of T ˇ to Λ 3 π * ⟨ ξ 2 , ξ 3 ⟩ , Λ 2 π * ⟨ ξ 2 , ξ 3 ⟩ ∧ π * ℋ and Λ 3 π * ℋ all vanish. Nagy [17] distinguishes types of nearly Kähler structures with reducible characteristic holonomy T N = V ˇ ⊕ ℋ ˇ via eigenvalues of the endomorphism F : ℋ ˇ → ℋ ˇ defined by

F ≔ − ∑ ( ∇ e i g N J ) 2 ,

where the sum is taken over a basis { e i } of V ˇ . In the situation at hand, V ˇ = π * ⟨ ξ 2 , ξ 3 ⟩ , ℋ ˇ = π * ℋ , and

F = − ∑ i = 2,3 ( ∇ ξ i ∘ s g N J ) 2 = − ∑ i = 2,3 ( ( ξ i ∘ s ) _ ∣ T ˇ ) 2 = − 4 α 2 ( φ 2 2 + φ 3 2 ) = 8 α 2 id ℋ ,

where we used ( ∇ J ) J = − J ( ∇ J ) and (10).

In his classification of nearly Kähler manifolds, Nagy [17] shows that such a manifold, if complete, is either homogeneous of type 3 in his notation or the twistor space of a quaternionic Kähler manifold. We prove with canonical submersions a local version similar to [4, Corollary 7.7].

Theorem 4.5

Let ( N , g N , J ) be a nearly Kähler manifold with characteristic connection ∇ T N . Assume the tangent space T N = V ˇ ⊕ ℋ ˇ splits into Hol 0 ( ∇ T N ) - and J-invariant subsets such that the characteristic torsion satisfies T N ∈ Λ 2 ℋ ˇ ∧ V ˇ . Then, there is a locally defined Riemannian submersion π : N → N ˇ along V ˇ . Furthermore, if F = k id ℋ ˇ , k > 0 , and dim V ˇ = 2 , then N ˇ admits a quaternionic Kähler structure locally defined by

I 1 = π * ∘ J ∘ s * , I 2 = 2 k π * ∘ ( J V _ ∣ T N ) ∘ s * , I 3 = 2 k π * ∘ ( V _ ∣ T N ) ∘ s * ,

for any section s : N ˇ → N of π and vertical vector field V ∈ Γ V ˇ of norm 1.

Remark 4.6

As Hol 0 ( ∇ T N ) ⊂ U ( n ) , the splitting T N = V ˇ 2 ⊕ ℋ ˇ 2 n − 2 into J - and Hol 0 ( ∇ T N ) -invariant modules is equivalent to Hol 0 ( ∇ T N ) ⊆ U ( 1 ) × U ( n − 1 ) with its standard representation on TN. The result in [4] used this assumption and the following assumption on the complex type

T N ∈ Λ 2,0 ℋ ˇ ∧ Λ 1,0 V ˇ ⊕ Λ 0,2 ℋ ˇ ∧ Λ 0,1 V ˇ .

Proof

( N , g N , ∇ T N ) satisfies the conditions in Theorem 2.1, so we obtain a locally defined Riemannian submersion π : N → N ˇ with totally geodesic fibers tangent to V such that

∇ X g N ˇ Y = π * ( ∇ X ¯ T N Y ¯ ) .

We check that I 1 , I 2 , and I 3 satisfy the quaternion relations. As in Theorem 4.1, we see immediately I 1 2 = − id . Observe that

( ∇ J V g N J ) 2 = ( ∇ V g N J ) J ( ∇ V g N J ) J = − ( ∇ V g N J ) 2 J 2 = ( ∇ V g N J ) 2 = − 1 2 F = − k 2 id

since V and J V form an orthonormal base of V ˇ . It follows that

(14) ( V _ ∣ T N ) 2 = ( ( ∇ V g N J ) J ) 2 = − k 2 id

and analogously for ( J V _ ∣ T N ) 2 . Therefore, I 2 and I 3 are almost complex structures on N ˇ . The quaternionic relations follow immediately from

(15) J ( V _ ∣ T N ) = J ( ∇ V g N J ) J = − ( ∇ J V g N J ) J = − ( J V _ ∣ T N )

and (14).

It remains to show that ∇ g N ˇ preserves the subbundle of End ( T N ˇ ) generated by I 1 , I 2 , and I 3 . We proceed as in Theorem 4.1 and set X ˆ ≔ X ¯ − s * X . Then,

( ∇ X g N ˇ I 1 ) Y = ∇ X g N ˇ ( I 1 Y ) − I 1 ( ∇ X g N ˇ Y ) = π * ( ∇ X ¯ T N ( π * ( J ( s * Y ) ) ) ¯ − J ( s * ( π * ( ∇ X ¯ T N Y ¯ ) ) ) ) = π * ( ∇ s * X T N ( J ( s * Y ) ) ) + ∇ X ˆ T N ( π * ( J ( s * Y ) ) ) ¯ − J ( ∇ s * X T N Y ¯ ) − J ( s * ( π * ( ∇ X ˆ T N Y ¯ ) ) ) = π * ( ( ∇ s * X T N J ) ( s * Y ) + ( X ˆ _ ∣ T N ) ( J ( s * Y ) ) − J ( X ˆ _ ∣ T N ) ( s * Y ) ) = 2 π * ( ( J X ˆ _ ∣ T N ) ( s * Y ) ) ,

where we made use of Lemma 2.5. This shows ∇ g N ˇ I 1 ∈ ⟨ I 2 , I 3 ⟩ since X ˆ ∈ V ˇ = ⟨ V , J V ⟩ . We play the game once more for I 3 . Then, the covariant derivative of I 2 is computed completely analogously:

( ∇ X g N ˇ I 3 ) Y = ∇ X g N ˇ ( I 3 Y ) − I 3 ∇ X g N ˇ Y = 2 k π * ( ∇ X ¯ T N ( π * ( ( V _ ∣ T N ) ( s * Y ) ) ) ¯ − ( V _ ∣ T N ) ( s * ( π * ( ∇ X ¯ T N Y ¯ ) ) ) ) = 2 k π * ( ∇ s * X T N ( ( V _ ∣ T N ) ( s * Y ) ) + ∇ X ˆ T N ( π * ( ( V _ ∣ T N ) ( s * Y ) ) ) ¯ − ( V _ ∣ T N ) ( ∇ s * X T N Y ¯ ) − ( V _ ∣ T N ) ( s * ( π * ( ∇ X ˆ T N Y ¯ ) ) ) ) = 2 k π * ( ( ∇ s * X T N ( V _ ∣ T N ) ) ( s * Y ) + ( X ˆ _ ∣ T N ) ( V _ ∣ T N ) ( s * Y ) − ( V _ ∣ T N ) ( X ˆ _ ∣ T N ) ( s * Y ) ) = 2 k π * ( ( ( ∇ s * X T N V ) _ ∣ T N ) ( s * Y ) ) + 2 k g N ( J V , X ˆ ) I 1 Y ,

where in the last step, we have used (14) and (15) to conclude

( X ˆ _ ∣ T N ) ( V _ ∣ T N ) − ( V _ ∣ T N ) ( X ˆ _ ∣ T N ) = g ( V , X ˆ ) ( ( V _ ∣ T N ) 2 − ( V _ ∣ T N ) 2 ) + g ( J V , X ˆ ) ( ( J V _ ∣ T N ) ( V _ ∣ T N ) − ( V _ ∣ T N ) ( J V _ ∣ T N ) ) = − g ( J V , X ˆ ) ( J ( V _ ∣ T N ) 2 + ( V _ ∣ T N ) 2 J ) = k g ( J V , X ˆ ) J .

Now, the result follows as ∇ s * X T N V ∈ V ˇ = ⟨ V , J V ⟩ .□

Acknowledgments

I would like to thank Giovanni Russo for many helpful discussions on the topic.

Funding information: The author states no funding involved.
Author contributions: The author confirms sole responsibility for the material and manuscript.
Conflict of interest: The author states no conflict of interest.

References

[1] I. Agricola, The Srní lectures on non-integrable geometries with torsion, Arch. Math. (Brno) 42 (2006), no. suppl., 5–84, https://doi.org/10.48550/arXiv.math/0606705. Search in Google Scholar

[2] I. Agricola and G. Dileo, Generalizations of 3-Sasakian manifolds and skew torsion, Adv. Geom. 20 (2020), no. 3, 331–374, https://doi.org/10.1515/advgeom-2018-0036. Search in Google Scholar

[3] I. Agricola, G. Dileo, and L. Stecker, Homogeneous non-degenerate 3-(α,δ)-Sasaki manifolds and submersions over quaternionic Kähler spaces, Ann. Global Anal. Geom. 60 (2021), no. 1, 111–141, https://doi.org/10.1007/s10455-021-09762-9. Search in Google Scholar

[4] B. Alexandrov, Sp(n)U(1)-connections with parallel totally skew-symmetric torsion, J. Geom. Phys. 57 (2006), no. 1, 323–337, https://doi.org/10.1016/j.geomphys.2006.03.005. Search in Google Scholar

[5] F. Belgun and A. Moroianu, Nearly Kähler 6-manifolds with reduced holonomy, Ann. Global Anal. Geom. 19 (2001), no. 4, 307–319, https://doi.org/10.1023/A:1010799215310. Search in Google Scholar

[6] R. L. Bryant, Metrics with exceptional holonomy, Ann. Math. 126 (1987), no. 2, 525–576, https://doi.org/10.2307/1971360. Search in Google Scholar

[7] R. Cleyton, A. Moroianu, and U. Semmelmann, Metric connections with parallel skew-symmetric torsion, Adv. Math. 378 (2021), 51, Id/No 107519, https://doi.org/10.1016/j.aim.2020.107519. Search in Google Scholar

[8] J. Eells and S. Salamon, Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12 (1985), 589–640, https://www.numdam.org/item/ASNSP_1985_4_12_4_589_0.pdf. Search in Google Scholar

[9] T. Friedrich and R. Grunewald, On Einstein metrics on the twistor space of a four-dimensional Riemannian manifold, Math. Nachr. 123 (1985), 55–60, https://doi.org/10.1002/MANA.19851230106. Search in Google Scholar

[10] T. Friedrich, Weak Spin(9)-structures on 16-dimensional Riemannian manifolds, Asian J. Math. 5 (2001), no. 1, 129–160, https://doi.org/10.48550/arXiv.math/9912112. Search in Google Scholar

[11] T. Friedrich, Spin(9)-structures and connections with totally skew-symmetric torsion, J. Geom. Phys. 47 (2003), no. 2–3, 197–206, https://doi.org/10.1016/S0393-0440(02)00189-4. Search in Google Scholar

[12] T. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2002), no. 2, 303–335, https://doi.org/10.48550/arXiv.math/0102142. Search in Google Scholar

[13] J. P. Gauntlett, D. Martelli, and D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D 69 (2004), 086002, https://doi.org/10.1103/PhysRevD.69.086002. Search in Google Scholar

[14] A. Gray, Weak holonomy groups, Math. Z. 123 (1971), 290–300, https://doi.org/10.1007/BF01109983. Search in Google Scholar

[15] A. Gray, The structure of nearly Kähler manifolds, Math. Ann. 223 (1976), 233–248, https://doi.org/10.1007/BF01360955. Search in Google Scholar

[16] A. Moroianu and P. Schwahn, Submersion constructions for geometries with parallel skew torsion, 2024, https://arxiv.org/abs/2409.14421. Search in Google Scholar

[17] P.-A. Nagy, Nearly Kähler geometry and Riemannian foliations, Asian J. Math. 6 (2002), no. 3, 481–504, https://doi.org/10.48550/arXiv.math/0203038. Search in Google Scholar

[18] L. Stecker, On 3-(α,δ)-sasaki manifolds and their canonical submersions, Ph.D. thesis, Philipps-Universität Marburg, August 2021, https://doi.org/10.17192/z2022.0071. Search in Google Scholar

Received: 2025-01-14

Revised: 2025-05-27

Accepted: 2025-06-26

Published Online: 2025-08-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/coma-2025-0016

Keywords for this article

skew-torsion connection; Riemannian submersions; 3-(α, δ)-Sasaki manifolds; nearly Kähler geometry; quaternionic Kähler manifolds

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