[Paper Review] Warped Product Pointwise Semi-slant Submanifolds of Sasakian Manifolds

Ion Mihai, Siraj Uddin|arXiv (Cornell University)|Jun 14, 2017

Geometric Analysis and Curvature Flows9 citations

TL;DR

This paper investigates warped product pointwise semi-slant submanifolds within Sasakian manifolds, extending recent work on pointwise slant submanifolds in almost Hermitian geometry. It provides a characterization theorem for such submanifolds and constructs non-trivial examples, establishing foundational geometric properties through differential geometric techniques.

ABSTRACT

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using this notion, we investigate pointwise semi-slant submanifolds and their warped products in Sasakian manifolds. We give non-trivial examples of such submanifolds and obtain several fundamental results, including a characterization for warped product pointwise semi-slant submanifolds of Sasakian manifolds.

Motivation & Objective

• To extend the theory of pointwise slant submanifolds to pointwise semi-slant submanifolds in Sasakian geometry.
• To study the geometric structure of warped product submanifolds where one factor is a pointwise semi-slant submanifold.
• To provide non-trivial examples of warped product pointwise semi-slant submanifolds in Sasakian manifolds.
• To derive a characterization theorem for warped product pointwise semi-slant submanifolds in Sasakian manifolds.
• To establish fundamental geometric properties and integrability conditions for such submanifolds.

Proposed method

• Adapting the notion of pointwise slant submanifolds from almost Hermitian geometry to Sasakian manifolds.
• Defining pointwise semi-slant submanifolds as a generalization of slant and totally real submanifolds in Sasakian settings.
• Constructing warped product structures using a Riemannian metric with a warping function on the base factor.
• Applying differential geometric tools, including the Gauss and Weingarten equations, to analyze the second fundamental form.
• Using the Sasakian structure's compatibility with the almost contact metric structure to derive integrability and curvature conditions.
• Deriving a characterization theorem based on the distribution of the slant distribution and the structure vector field.

Experimental results

Research questions

• RQ1 How can the concept of pointwise slant submanifolds be generalized to pointwise semi-slant submanifolds in Sasakian manifolds?
• RQ2 What are the geometric constraints and structural properties of warped product pointwise semi-slant submanifolds in Sasakian manifolds?
• RQ3 Can non-trivial examples of such submanifolds be explicitly constructed?
• RQ4 What conditions characterize warped product pointwise semi-slant submanifolds within Sasakian manifolds?
• RQ5 How does the warping function influence the geometry of the submanifold and its integrability?

Key findings

• A complete characterization theorem is established for warped product pointwise semi-slant submanifolds in Sasakian manifolds.
• Non-trivial examples of warped product pointwise semi-slant submanifolds are constructed, demonstrating the existence of such geometric structures.
• The paper confirms that the slant distribution and the structure vector field play a crucial role in determining the integrability and curvature properties.
• The second fundamental form of the submanifold satisfies specific relations derived from the Sasakian structure and the warping function.
• The geometry of the submanifold is shown to be constrained by the interplay between the warping function and the pointwise slant angle distribution.
• The results generalize previous findings on pointwise slant submanifolds and extend the theory of warped products in contact geometry.