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On semi-symmetric recurrent-metric connection. (English) Zbl 0817.53024

The semi-symmetric recurrent-metric connections on a Riemannian manifold was introduced by O. C. Andonie and D. Smaranda [Tensor, New Ser. 31, 8-12 (1977; Zbl 0369.53017)]. This paper is a continuation of the author’s previous one [J. Xiamen Univ., Nat. Sci. 27, No. 3, 248-253 (1988)]. Let \((M,g)\) be an \(n\)-dimensional Riemannian manifold \((n > 2)\) and \(D\) a semi-symmetric connection on \(M\). If \(D\) satisfies the condition \(D_ X g = 2\mu(X)g\), where \(\mu\) is a linear differential form and \(X\) is a vector field on \(M\), then \(D\) is said to be a semi- symmetric recurrent-metric connection. The author investigates the relations between semi-symmetric recurrent-metric connections and the Riemannian connection of \(M\). The case of isotropic semi-symmetric recurrent-metric connections is also studied.
Theorem: A semi-symmetric recurrent-metric connection \(D\) on \(M\) is conformally flat if and only if \(M\) is conformally flat and the recurrent factor \(\mu(\mu_ i)\) of \(D\) is a gradient vector.

MSC:

53C20 Global Riemannian geometry, including pinching
53B20 Local Riemannian geometry
53B05 Linear and affine connections
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53B21 Methods of local Riemannian geometry

Citations:

Zbl 0369.53017
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