Generalized Sasakian space forms and Riemannian manifolds of quasi constant sectional curvature

TL;DR

This paper classifies generalized Sasakian space forms and Riemannian manifolds with quasi-constant sectional curvature, revealing their structure as either of constant curvature, certain hypersurfaces, or specific product manifolds.

Contribution

It provides a comprehensive local classification of higher-dimensional generalized Sasakian space forms and Riemannian manifolds with quasi-constant sectional curvature.

Findings

Classifies generalized Sasakian space forms of dimension >5.

Identifies conditions under which these forms are of constant curvature or specific product types.

Provides a local classification for Riemannian manifolds of quasi-constant sectional curvature.

Abstract

In this paper, we show that a generalized Sasakian space form of dimension greater than three is either of constant sectional curvature; or a canal hypersurface in Euclidean or Minkowski spaces; or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold; or locally a wapred product of a real line and a generalized complex space form; or an $\alpha$-Sasakian space form; or it is of five dimension and admits an $\alpha$-Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension greater than five is obtained. A local classification of Riemannian manifolds of quasi constant sectional curvature of dimension greater than three is also given in this paper.