[Paper Review] Anti-holomorphic semi-invariant submersions from Kählerian manifolds

Hakan Mete Taştan|arXiv (Cornell University)|Apr 9, 2014

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper investigates anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds, proving integrability of the involved distributions and vanishing of the O’Neill tensor 𝒯 on the invariant vertical distribution. It establishes necessary and sufficient conditions for harmonicity and total geodesicness, derives explicit curvature formulas relating the total space, fibers, and base, and proves that no proper such submersion exists from a non-flat complex space form (c ≠ 0).

ABSTRACT

We study anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds. We prove that all distributions which are involved in the definition of the submersion are integrable. We also prove that the O'Neill's tensor $\mathcal{T}$ vanishes on the invariant vertical distribution. We give necessary and sufficient conditions for totally geodesicness and harmonicity of this type submersions. Moreover, we investigate the several curvatures of the total manifold and fibers and give a characterization theorem.

Motivation & Objective

• To investigate the geometric properties of anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds.
• To determine conditions under which such submersions are harmonic or totally geodesic.
• To derive curvature formulas relating the total space, fibers, and base manifold.
• To characterize the geometry of the total manifold and fibers using curvature invariants.
• To establish a non-existence result for proper submersions from non-flat complex space forms.

Proposed method

• Utilizes Riemannian submersion theory with O’Neill’s tensors 𝒯 and 𝒜 to analyze geometric structures.
• Applies the Kähler condition and anti-holomorphic semi-invariant distribution decomposition: 𝒟 and 𝒟⊥.
• Derives curvature formulas using O’Neill’s curvature identities and the Kähler condition.
• Employs the O’Neill tensor 𝒯 to measure vertical curvature and integrability obstructions.
• Uses the Levi-Civita connection and pullback connections to compute second fundamental forms.
• Applies Lemma 5.5 and Theorem 5.7 to simplify curvature expressions involving ∇𝒯.

Experimental results

Research questions

• RQ1 Under what conditions are the horizontal and vertical distributions of an anti-holomorphic semi-invariant submersion integrable?
• RQ2 When does the O’Neill tensor 𝒯 vanish on the invariant vertical distribution?
• RQ3 What are the necessary and sufficient conditions for such submersions to be harmonic or totally geodesic?
• RQ4 How do the sectional and holomorphic bisectional curvatures of the total space relate to those of the base and fibers?
• RQ5 Can proper anti-holomorphic semi-invariant submersions exist from non-flat complex space forms (c ≠ 0)?

Key findings

• The horizontal and vertical distributions of an anti-holomorphic semi-invariant submersion from a Kählerian manifold are integrable.
• The O’Neill tensor 𝒯 vanishes on the invariant vertical distribution, implying no vertical curvature obstruction there.
• The submersion is totally geodesic if and only if the horizontal distribution is totally geodesic and 𝒯 vanishes on the horizontal distribution.
• The submersion is harmonic if and only if the mean curvature vector of the fibers vanishes and the horizontal distribution is totally geodesic.
• Sectional curvature formulas (7.6)–(7.11) explicitly relate the curvature of the total space to the base, fibers, and 𝒯-tensor.
• No proper anti-holomorphic semi-invariant submersion exists from a complex space form with c ≠ 0; the total space must be flat (c = 0).