[Paper Review] Geometry of Slant Submanifolds

Bang‐Yen Chen|arXiv (Cornell University)|Jan 1, 1990

Geometric Analysis and Curvature Flows21 references263 citations

TL;DR

This paper systematically presents foundational work on slant submanifolds in Riemannian geometry, developed through lectures at KU Leuven in 1990. It introduces the geometric classification of submanifolds via slant angles, establishes intrinsic and extrinsic properties, and derives key curvature relations, contributing a unified framework for studying slant geometry in complex and almost complex manifolds.

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 provide a systematic exposition of the geometry of slant submanifolds based on original research and collaborative work.
• To clarify the role of slant angles in classifying submanifolds within almost complex and complex manifolds.
• To establish intrinsic and extrinsic geometric properties of slant submanifolds, including curvature relations.
• To unify and extend previous results on slant submanifolds through a coherent theoretical framework.
• To explore connections between slant geometry and related topics in differential geometry, including complex structures and curvature tensor properties.

Proposed method

• Leveraging the concept of slant angle to classify submanifolds in almost Hermitian manifolds.
• Applying differential geometric techniques to analyze the second fundamental form and shape operator of slant submanifolds.
• Using the Gauss and Codazzi equations to derive curvature relations for slant submanifolds.
• Integrating results from joint work with Nagano, Tazawa, Leung, and Morvan to generalize known theorems.
• Formulating geometric conditions under which submanifolds are minimal or totally geodesic in the context of slant geometry.
• Employing a systematic lecture-based approach to organize and present results in a coherent, accessible manner.

Experimental results

Research questions

• RQ1 How can slant submanifolds be systematically classified using the angle between tangent spaces and almost complex structures?
• RQ2 What are the intrinsic and extrinsic geometric properties of slant submanifolds in almost Hermitian manifolds?
• RQ3 What curvature relations arise from the Gauss and Codazzi equations in the context of slant submanifolds?
• RQ4 Under what conditions are slant submanifolds minimal or totally geodesic?
• RQ5 How do the joint results with Nagano, Tazawa, Leung, and Morvan extend the theory of slant submanifolds?

Key findings

• The geometry of slant submanifolds is fully characterized by the slant angle, which determines the angle between the tangent space and the almost complex structure.
• The second fundamental form of a slant submanifold satisfies specific symmetry and curvature constraints derived from the slant condition.
• Curvature relations between the ambient manifold and the submanifold are established via the Gauss and Codazzi equations applied to slant submanifolds.
• Minimal slant submanifolds arise under specific geometric conditions involving the mean curvature vector and the slant angle.
• Totally geodesic slant submanifolds exist only when the slant angle is zero or π/2, corresponding to complex or totally real submanifolds.
• The framework unifies and generalizes earlier results on slant submanifolds, providing a coherent foundation for further research in complex and almost complex geometry.