$ N(κ)- $contact Metric Manifolds with Generalized Tanaka-Webster Connection

TL;DR

This work investigates curvature properties of $N(\kappa)$-contact metric manifolds endowed with a generalized Tanaka-Webster connection. It derives curvature relations between the Levi-Civita connection and the generalized connection, and proves that when the manifold is K-contact it becomes a generalized Sasakian space form with specific $F_i$ values; it also analyzes concircular curvature tensors to establish eta-Einstein conditions under certain flatness assumptions. An explicit 3-dimensional example supports the theory and demonstrates the (non)existence of particular concircular-flatness conditions in this generalized setting. Overall, the paper clarifies rigidity and curvature structure in contact geometry under the generalized Tanaka-Webster framework and highlights when such manifolds resemble Sasakian-space-forms.

Abstract

In this paper we work on $N(κ)$-contact metric manifolds with a generalized Tanaka-Webster connection . We obtain some curvature properties. It is proven that if a $N(κ)$-contact metric manifold with generalized Tanaka-Webster connection is K-contact then it is an example of generalized Sasakian space form. Also we examine some flatness and symmetric conditions of concircular curvature tensor on a $N(κ)$-contact metric manifolds with a generalized Tanaka-Webster connection.

Theorems & Definitions (18)

• Lemma 1
• Example 1
• Example 2
• Definition 1: tanno1989variational
• Lemma 2
• Corollary 1
• Remark 1
• Theorem 1
• proof
• Corollary 2