Curvature Properties and 𝜂-Einstein (𝑘,𝜇)-Contact Metric Manifolds

Table 1

Comparison of the results for different curvature tensors in 𝑀(𝑘,𝜇).

Curvature tensor	Condition	Result
𝐶(𝑋,𝑌,𝑍,𝑊)	(∇𝑊𝐶)(𝑋,𝑌)𝑍=0	𝑀(𝑘,𝜇) is cosymplectic ⇔; 𝜇=2(1−𝑛)⇔; 𝑀(𝑘,𝜇) is 𝜂-Einstein
𝐶(𝑋,𝑌,𝑍,𝑊)	𝑅⋅𝐶=0	⇔𝜂-Einstein with 𝛼=−2𝑛𝑘 and 𝛽=4𝑛𝑘𝜇(𝑘+𝜇)
𝐶(𝑋,𝑌,𝑍,𝑊)	𝐶(𝜙𝑋,𝜙𝑌,𝜙𝑍,𝜙𝑊)=0	Ricci tensor is 𝜂-parallel and 𝜇=2(𝑛−1)𝑛
𝐶(𝑋,𝑌,𝑍,𝑊)	𝐶(𝑋,𝑌)𝜉=0	𝜂-Einstein with 𝛽=−𝛼−2𝑛𝑘, 𝛼=𝑘−(2𝑛−1)𝜇[2𝑛𝜇+2(𝑛−1)](𝑘−1)2(𝑛−1)−𝑛𝜇−𝑘
𝐶(𝑋,𝑌,𝑍,𝑊)	𝑀(𝑘,𝜇) is 𝜙-recurrent	𝜂-Einstein with 𝛽=2𝑛𝑘−𝛼, 𝛼=−𝑘+𝜇(2𝑛−1)(𝑘−1)𝑎+𝑘[𝑏−𝜇(2𝑛−1)]
𝐶(𝑋,𝑌)𝑍	𝐶(𝑋,𝑌)𝜉=0	𝑀(𝑘,𝜇) reduces to 𝑁(𝑘)-c.m.m ⇔𝑀(𝑘,𝜇) is 𝜂-Einstein
𝐶(𝑋,𝑌)𝑍	𝐶⋅𝑆=0	𝑀(1,−2) is 𝜂-Einstein with 𝛼=2𝑛(2𝑛−1)5𝑛−2, 𝛽=2𝑛(1−𝑛)5𝑛−2
𝐶(𝑋,𝑌)𝑍	′ 𝐶(𝜙𝑋,𝜙𝑌,𝜙𝑍,𝜙𝑊)=0	𝜂-Einstein with 𝛼=𝑟(4𝑛−1)+4𝑛^2𝑘2𝑛,𝛽=−𝑟(4𝑛−1)2𝑛
𝑃(𝑋,𝑌)𝑍	𝑅⋅𝑃=0	𝜂-Einstein
𝑃(𝑋,𝑌)𝑍	𝑃⋅𝑆=0	𝜂-Einstein
𝑅(𝑋,𝑌)𝑍	𝑅⋅𝑆=0	𝜂-Einstein