[Paper Review] Anti-invariant Riemannian maps from almost Hermitian manifolds

Bayram Şahin|arXiv (Cornell University)|Oct 1, 2012

Geometric Analysis and Curvature Flows14 references3 citations

TL;DR

This paper introduces anti-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds as a generalization of anti-invariant Riemannian submersions, establishing geometric properties of the induced foliations and proving a decomposition theorem for the source manifold. The key result is that every pluriharmonic Lagrangian Riemannian map from a Kähler manifold to a Riemannian manifold is totally geodesic.

ABSTRACT

As a generalization of anti-invariant Riemannian submersions, we introduce anti-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We give examples and investigate the geometry of foliations which are arisen from the definition of an anti-Riemannian map. Then we give a decomposition theorem for the source manifold of such maps. We also find necessary and sufficient conditions for anti-invariant Riemannian maps to be totally geodesic and show that every pluriharmonic Lagrangian Riemannian map, which is a special anti-invariant Riemannian map, from a Kähler manifold to a Riemannian manifold is totally geodesic.

Motivation & Objective

• To generalize anti-invariant Riemannian submersions by introducing anti-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds.
• To investigate the geometry of foliations induced by the kernel and horizontal distributions of such maps.
• To establish a decomposition theorem for the source manifold under anti-invariant Riemannian maps.
• To derive necessary and sufficient conditions for such maps to be totally geodesic.
• To prove that pluriharmonic Lagrangian Riemannian maps from Kähler manifolds to Riemannian manifolds are totally geodesic.

Proposed method

• Define anti-invariant Riemannian maps as smooth maps $ F: (M, g_M, J) \to (N, g_N) $ where $ J(ker F_*) \subseteq (ker F_*)^\perp $, generalizing anti-invariant submersions.
• Use the decomposition $ TM = ker F_* \oplus \mathcal{H} $ with $ \mathcal{H} = (ker F_*)^\perp $, and $ TN = range F_* \oplus (range F_*)^\perp $, ensuring $ F $ is a subimmersion with constant rank.
• Apply the condition that the horizontal restriction $ F^h_*: \mathcal{H} \to range F_* $ is a linear isometry, satisfying $ g_M(X,Y) = g_N(F_*X, F_*Y) $ for $ X,Y \in \mathcal{H} $.
• Analyze the second fundamental form $ \nabla F_* $ and use the pluriharmonicity condition $ (\nabla F_*)(X,Y) + (\nabla F_*)(JX,JY) = 0 $ for $ X,Y \in \Gamma(ker F_*) $.
• Use integrability of $ ker F_* $ via the rank theorem and Kähler structure to show $ \nabla F_* $ vanishes on $ ker F_* \times \mathcal{H} $, implying total geodesy.
• Apply the twisted product decomposition theorem to show that if $ M_1 $ is locally a twisted product $ M_{(ker F_*)^\perp} \times_f M_{ker F_*} $, then no such Lagrangian Riemannian map exists.

Experimental results

Research questions

• RQ1 What is the geometric structure of the foliations induced by the kernel and horizontal distributions of an anti-invariant Riemannian map?
• RQ2 Under what conditions can the source manifold of an anti-invariant Riemannian map be decomposed into a twisted product?
• RQ3 When is an anti-invariant Riemannian map totally geodesic?
• RQ4 Is every pluriharmonic Lagrangian Riemannian map from a Kähler manifold to a Riemannian manifold necessarily totally geodesic?
• RQ5 Can a Lagrangian Riemannian map exist on a locally twisted product manifold of the form $ M_{(ker F_*)^\perp} \times_f M_{ker F_*} $?

Key findings

• Anti-invariant Riemannian maps generalize anti-invariant Riemannian submersions by allowing the range of $ F_* $ to be a proper subspace of $ TN $, with $ (range F_*)^\perp \neq \{0\} $.
• The source manifold of an anti-invariant Riemannian map admits a decomposition into a twisted product $ M_{(ker F_*)^\perp} \times_f M_{ker F_*} $ if the horizontal and vertical distributions are integrable and satisfy curvature conditions.
• A necessary and sufficient condition for an anti-invariant Riemannian map to be totally geodesic is that the second fundamental form $ \nabla F_* $ vanishes on $ ker F_* \times ker F_* $.
• Every pluriharmonic Lagrangian Riemannian map from a Kähler manifold to a Riemannian manifold is totally geodesic, as shown by the vanishing of $ \nabla F_* $ on $ ker F_* \times ker F_* $, $ ker F_* \times \mathcal{H} $, and $ \mathcal{H} \times ker F_* $.
• There does not exist a Lagrangian Riemannian map from a Kähler manifold to a Riemannian manifold if the source is locally isometric to a twisted product $ M_{(ker F_*)^\perp} \times_f M_{ker F_*} $, due to integrability and curvature obstructions.
• The integrability of $ ker F_* $ is guaranteed by the rank theorem and the fact that $ F $ is a subimmersion with constant rank, which is essential for the decomposition and geodesy results.