Quasi-isometry between two almost contact metric manifolds

TL;DR

The paper introduces and analyzes quasi-isometry for almost contact metric manifolds, with a focus on $N(k)$-contact and Sasakian structures. It develops a framework of curvature-compatibility inequalities under quasi-isometric embeddings, derives η-Einstein and related curvature consequences under conformal-type flatness assumptions, and provides an explicit Sasakian example illustrating the construction. A general curvature-comparison scheme is also established for arbitrary Riemannian manifolds under quasi-isometries, linking scalar and Ricci curvatures to the quasi-isometric constants. Overall, the work advances understanding of how geometric structure and curvature behave under quasi-isometries in odd-dimensional contact geometry, offering tools for rigidity and comparison results.

Abstract

In this paper the notion of quasi-isometry between two Riemannian manifolds has been introduced. This idea is also imposed to study quasi-isometry between two almost contact metric manifolds. Moving further, some curvature properties of two quasi-isometrically embedded almost contact metric manifolds, $N(k)-$contact metric manifolds and Sasakian manifolds are investigated. Next, an illustrative example of a quasi-isometry between two Sasakian structures is constructed. Finally, a relation between the scalar curvature and the quasi-isometric constants for two quasi-isometric Riemannian manifolds has been established.

Theorems & Definitions (19)

• Definition 1.1
• Proposition 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 2.4
• Proposition 2.5
• Theorem 3.1
• Theorem 4.1
• Remark 4.2
• Corollary 4.3