Abstract
The object of the present paper is to study semi-symmetric metric connection on a 3-dimensional trans-Sasakian manifold. We found the necessary condition under which a vector field on a 3-dimensional trans-Sasakian manifold will be a strict contact vector field. Then, we obtained extended generalized phi-recurrent 3-dimensional trans-Sasakian manifold with respect to semi-symmetric metric connection. Next, a 3-dimensional trans-Sasakian manifold satises the condition ~L.~ S = 0 with respect to semi-symmetric metric connection is studied.
Highlights
The semi symmetric metric connection gives an important concept in the geometry of Riemannian manifold having physical applications i.e the displacement on the earth surface facing a fixed point is metric and semi-symmetric [13]
We have considered that M 3 is 3-dimensional trans-Sasakian manifold
An extended generalized φ-recurrent trans-Sasakian manifold (M 3, g) with respect to semi symmetric metric connection is an η− Einstein manifold and more over, the 1-forms A and B are related as A(W )[α2 − β2 − β] + B(W ) = 0
Summary
A three-dimensional trans-Sasakian manifold is said to be an extended generalized φ−recurrent trans-Sasakian manifold if its curvature tensor R satisfies the relation φ2((∇W R)(X, Y )Z) = A(W )φ2(R(X, Y )Z) + B(W )φ2([g(Y, Z)X − g(X, Z)Y ]), for all X, Y, Z, W ∈ χ(M ), where A and B are two non-vanishing 1-forms such that g(W, ρ1) = A(W ) and g(W, ρ2) = B(W ), ∀W ∈ χ(M ), with ρ1 and ρ2 being the vector fields associated to the 1-form A and B, respectively [16]. Contracting (4.5) with gik, we get Rjk,k = a,j + b,kξkηj + bηi,kgikηj + bηiηj,kgik. An extended generalized φ-recurrent trans-Sasakian manifold (M 3, g) with respect to semi symmetric metric connection is an η− Einstein manifold and more over, the 1-forms A and B are related as A(W )[α2 − β2 − β] + B(W ) = 0.
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