[Paper Review] A survey on Geometry of Slant Submanifolds

Bang‐Yen Chen|arXiv (Cornell University)|Jul 5, 2013

Geometric Analysis and Curvature Flows25 references3 citations

TL;DR

This survey systematically develops the geometry of slant submanifolds in almost Hermitian and Kaehlerian manifolds, introducing them as a natural generalization between complex and totally real submanifolds. It establishes foundational theory, classifies slant surfaces via curvature and topology, and proves key results on stability and cohomology, notably the instability of minimal totally real totally geodesic submanifolds in complex Grassmannians.

ABSTRACT

The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.

Motivation & Objective

• To present a systematic theory of slant submanifolds as a generalization of complex and totally real submanifolds in almost Hermitian and Kaehlerian manifolds.
• To classify slant surfaces based on geometric properties such as parallel mean curvature vector, sphericality, and codimension.
• To investigate the topology and stability of slant submanifolds, particularly focusing on minimal and totally geodesic cases.
• To establish cohomological invariants for slant submanifolds and relate them to geometric stability.
• To provide a comprehensive overview of recent results and open problems in the geometry of slant submanifolds, drawing on joint work with Nagano, Tazawa, Leung, and Morvan.

Proposed method

• Introduces slant submanifolds via the angle between the almost complex structure J and tangent spaces, generalizing complex (angle 0) and totally real (angle π/2) submanifolds.
• Uses the Gauss map and geometry of Grassmannian manifolds G(2,4) to analyze slant surfaces in C² and almost Hermitian manifolds.
• Applies representation theory and Casimir operators on symmetric spaces to compute index, nullity, and Killing nullity of totally geodesic submanifolds.
• Employs Schur’s lemma and orthogonal decompositions of Lie algebras to analyze irreducible components of normal bundles.
• Utilizes cohomological tools from [CM3] and [CLN] to study the cohomology of α-oblique submanifolds.
• Applies Ohnita’s formulas for index, nullity, and Killing nullity to compact totally geodesic submanifolds in symmetric spaces.

Experimental results

Research questions

• RQ1 How can slant submanifolds be characterized geometrically in terms of the angle between J and tangent spaces?
• RQ2 What are the classification criteria for slant surfaces in terms of curvature, codimension, and mean curvature?
• RQ3 Under what conditions are slant submanifolds stable or unstable as critical points of the volume functional?
• RQ4 How do cohomological invariants such as H^1 and H^2 relate to the topology and geometry of slant submanifolds?
• RQ5 What is the precise relationship between the Casimir eigenvalues of the normal bundle and the stability of totally geodesic submanifolds?

Key findings

• Minimal totally real totally geodesic submanifolds such as G^R(p,q) in G^C(p,q) are unstable, as shown by comparing Casimir eigenvalues c(2ω̃₁) > c(ω̃₂).
• The index, nullity, and Killing nullity of compact totally geodesic submanifolds in symmetric spaces are computed via Ohnita’s formulas using representation theory.
• Slant surfaces with rk(ν) < 2 or codimension one admit specific classification results, including topological constraints and curvature conditions.
• Spherical slant surfaces are shown to be minimal and have restricted topology, with all M₊ homologous to zero.
• The cohomology of slant submanifolds is studied via the complex of α-oblique forms, with H^1 and H^2 providing topological invariants.
• The Gauss map of a slant surface in C² is shown to be harmonic if and only if the surface is minimal and slant, linking geometry to analysis.