[Paper Review] Semi-invariant submersions from almost Hermitian manifolds

Bayram Şahin|arXiv (Cornell University)|Jun 1, 2010

Geometric Analysis and Curvature Flows8 references3 citations

TL;DR

This paper introduces semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds, generalizing both anti-invariant and almost Hermitian submersions. It establishes geometric conditions for the total manifold to be locally a product, characterizes totally geodesic submersions, and proves that semi-invariant submersions with totally umbilical fibers on complex space forms must have zero curvature (c=0), with a classification showing fibers are either totally geodesic or the horizontal distribution is one-dimensional.

ABSTRACT

We introduce semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and find necessary-sufficient conditions for total manifold to be locally product Riemannian manifold. We also find necessary and sufficient conditions for a semi-invariant submersion to be totally geodesic. Moreover, we obtain a classification for semi-invariant submersions with totally umbilical fibers and show that such submersions put some restrictions on total manifolds.

Motivation & Objective

• To generalize anti-invariant and almost Hermitian submersions by introducing semi-invariant Riemannian submersions from almost Hermitian manifolds.
• To investigate the geometry of the foliations induced by the submersion, particularly the integrability and geometric structure of the leaves.
• To derive necessary and sufficient conditions for the total manifold to be locally a product Riemannian manifold.
• To determine conditions under which such submersions are totally geodesic.
• To classify semi-invariant submersions with totally umbilical fibers and analyze their curvature constraints.

Proposed method

• Define semi-invariant Riemannian submersions as Riemannian submersions from an almost Hermitian manifold where the vertical distribution is anti-invariant and the horizontal distribution is invariant under the almost complex structure.
• Use O’Neill’s tensors 𝒜 and 𝒯 to analyze the geometry of the submersion and derive curvature relations.
• Apply the curvature tensor of complex space forms to constrain the total manifold’s curvature when fibers are totally umbilical.
• Use the mean curvature vector field H and show it lies in J𝒟₂ using tensorial identities and the structure equations.
• Leverage the decomposition of the tangent bundle into distributions: kerF*, (kerF*)⊥, and the J-invariant subbundle 𝒟₂.
• Prove classification theorems by analyzing the vanishing of certain tensorial components and the orthogonality of distributions.

Experimental results

Research questions

• RQ1 Under what conditions is the total manifold of a semi-invariant submersion locally a product Riemannian manifold?
• RQ2 What are the necessary and sufficient conditions for a semi-invariant submersion to be totally geodesic?
• RQ3 What curvature constraints arise when a semi-invariant submersion has totally umbilical fibers on a complex space form?
• RQ4 How does the mean curvature vector field of the fibers behave in such submersions?
• RQ5 What classification can be achieved for semi-invariant submersions with totally umbilical fibers?

Key findings

• The total manifold is locally a product Riemannian manifold if and only if specific tensorial conditions involving the covariant derivative of F*, the tensors 𝒜 and 𝒯, and the distributions kerF* and (kerF*)⊥ vanish.
• A semi-invariant submersion is totally geodesic if and only if the tensors 𝒜 and 𝒯 vanish on the relevant distributions.
• If a semi-invariant submersion has totally umbilical fibers and the total manifold is a complex space form, then the curvature must be zero (c=0).
• The mean curvature vector field H of the fibers lies in the J-invariant subbundle J𝒟₂, which restricts the geometric structure of the fibers.
• Semi-invariant submersions with totally umbilical fibers on Kähler manifolds must have either one-dimensional horizontal distribution 𝒟₂ or totally geodesic fibers.