A Basic Inequality for the Tanaka‐Webster Connection

Dae Ho Jin,Jae Won Lee

doi:10.1155/2012/231416

Dae Ho Jin, Jae Won Lee

Open Access

https://doi.org/10.1155/2012/231416

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Abstract

For submanifolds tangent to the structure vector field in Sasakian space forms, we establish a Chen′s basic inequality between the main intrinsic invariants of the submanifold (namely, its pseudosectional curvature and pseudosectional curvature on one side) and the main extrinsic invariant (namely, squared pseudomean curvature on the other side) with respect to the Tanaka‐Webster connection. Moreover, involving the pseudo‐Ricci curvature and the squared pseudo‐mean curvature, we obtain a basic inequality for submanifolds of a Sasakian space form tangent to the structure vector field in terms of the Tanaka‐Webster connection.

Highlights

One of the basic interests in the submanifold theory is to establish simple relationship between intrinsic invariants and extrinsic invariants of a submanifold
We introduce pseudosectional curvatures and pseudo-Ricci curvature for the Tanaka-Webster connection in a Sasakian space form
We study basic inequalities for submanifolds of a Sasakian space form of a constant pseudosectional curvature and a pseudo-Ricci curvature in terms of the Tanaka-Webster connection

Summary

Introduction

One of the basic interests in the submanifold theory is to establish simple relationship between intrinsic invariants and extrinsic invariants of a submanifold. Chen 1 established a nice basic inequality-related intrinsic quantities and extrinsic ones of submanifolds in a space form with arbitrary codimension. He studied the basic inequalities of submanifolds of complex space forms and characterize submanifolds when the equality holds. We introduce pseudosectional curvatures and pseudo-Ricci curvature for the Tanaka-Webster connection in a Sasakian space form. We study basic inequalities for submanifolds of a Sasakian space form of a constant pseudosectional curvature and a pseudo-Ricci curvature in terms of the Tanaka-Webster connection

Preliminaries

The Tanaka-Webster Connection for Sasakian Space Form

A Pseudo-Ricci Curvature for Sasakian Space Form