On 3(α,δ)Sasaki Manifolds and their Canonical Submersions
We investigate 3(α,δ)Sasaki manifolds through their canonical connection. These manifolds are almost 3contact metric manifolds satisfying the condition dη_i = 2αΦ_i + 2(α − δ)η_j ∧ η_k, thus generalizing 3Sasaki manifolds. They admit a canonical metric connection ∇ with parallel skew torsion uni...
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Format:  Doctoral Thesis 
Language:  English 
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PhilippsUniversität Marburg
2021

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Summary:  We investigate 3(α,δ)Sasaki manifolds through their canonical connection. These manifolds are almost 3contact metric manifolds satisfying the condition dη_i = 2αΦ_i + 2(α − δ)η_j ∧ η_k, thus generalizing 3Sasaki manifolds. They admit a canonical metric connection ∇ with parallel skew torsion uniquely characterized by ∇_Xφ_i = β(η_k(X)φ_j − η_j(X)φ_k), β ∈ R. (1)
This condition restricts the holonomy hol(∇) ⊂ (sp(n)⊕sp(1))⊕so(3) and thereby induces a locally defined Riemannian submersion π along the Reeb orbits. This canonical submersion connects the geometries on the base N and total space by the key equation ∇^{g_N}_X Y = π_∗(∇_XY ). (2)
Combining (1) and (2), we obtain that for the projections φˇ_i of φ_i ∇^{g_N}_X φˇi = 2δ(ηˇ_k(X)φˇ_j − ηˇ_j(X)φˇ_k), defining a quaternionic Kähler structure on the base N. We show that the scalar curvature of N is given by scal_gN =16n(n+2)αδ, leading to vastly different behavior if αδ = 0, αδ > 0 or αδ < 0. These cases are called degenerate, positive and negative respectively.
Nondegenerate homogeneous 3(α, δ)Sasaki manifolds are a particularly wellbehaved class to investigate, since the underlying homogeneous quaternionic Kähler manifolds of nonvanishing scalar curvature are classified. The first option are symmetric spaces of compact (scal_gN > 0) and noncompact (scal_gN < 0) type. We treat this case in a unified manner showing that pairs of positive and negative homogeneous 3(α, δ)Sasaki manifolds appear as quotients of simple Lie groups. The other option are Alekseevsky spaces which admit a transitive solvable group action. These are harder to deal with than those admitting a transitive simple group action, but a careful construction gives rise to additional negative homogeneous 3(α, δ)Sasaki manifolds.
In a third part we exploit identity (2) once more applying it to investigate the curvature. This yields a decomposition of the canonical curvature operator R = αβR_⊥ + R_par where R_⊥ is controlled by the 3(α, δ)Sasaki structure and Rpar encodes the Riemannian curvature of the quaternionic Kähler base. If R_gN ≥ 0 or R_gN ≤ 0 and α, δ are suitable parameters, the decomposition allows us to derive the same properties for R. By the formula R_g = R + 1/4T + 1/4S_T we can also control positivity of R_g. These results are tailormade for the first class of homogeneous examples as their symmetric base automatically satisfies R_gN ≥ 0 or R_gN ≤ 0 depending on whether they are compact or not. We thus obtain semidefinite curvature operators on all of them and strongly positive curvature in some instances. 

Physical Description:  85 Pages 
DOI:  10.17192/z2022.0071 