Sasakian manifolds satisfying certain conditions on Q tensor
- PDF / 289,893 Bytes
- 10 Pages / 439.37 x 666.142 pts Page_size
- 42 Downloads / 163 Views
Journal of Geometry
Sasakian manifolds satisfying certain conditions on Q tensor H¨ ulya Ba˘gdatli Yilmaz Abstract. The object of the present work is to investigate Sasakian manifolds satisfying certain conditions on Q tensor whose trace is well-known Z tensor. Mathematics Subject Classification. Primary 53C25, Secondary 53D10. Keywords. Sasakian manifold, Q tensor, Unit sphere, Weyl conformal curvature tensor field, Projective curvature tensor field.
1. Introduction About two centuries ago, Lie [8] introduced the concept of contact transformation to study systems of differential equations from a geometrical point of view. This concept can be seen as the beginning of contact geometry. The subject has manifold connections with other fields of pure mathematics, and a significant place in applied areas like optics, control theory or mechanics, etc... Contact geometry has not been receiving attention for a long time until Chen [3] and Gray [5] contributed to its development. Later, Sasaki [14] introduced the notion of a Sasakian manifold which is a contact manifold with a Sasakian metric to be a special kind of Riemannian metric g. In other words, Sasaki introduced the concept of Sasakian structure as an odd-dimensional analogy of a K¨ ahler structure, [14]. Sasakian manifolds are a significant class of contact manifolds and studied by many authors (see, [1,4,6,7,12,22]). Let (M n , g) be an n(= 2m + 1)−dimensional Riemannian manifold of class C ∞ . Let there exist in M n a 1-form η, the associated vector field ξ and φ is a (1, 1)-tensor field such that φ2 = −X + η(X)ξ, n
(1.1)
for any vector field X. Then M is called an almost contact manifold and the system (φ, ξ, η) is called an almost contact structure to M n . 0123456789().: V,-vol
48
Page 2 of 10
H. B. Yilmaz
J. Geom.
From (1.1), it follows that [2] φξ = 0,
η(φX) = 0,
η(ξ) = 1.
(1.2)
If the Riemannian metric g in M n satisfies g(φX, φY ) = g(X, Y ) − η(X)η(Y ), n
(1.3)
n
for any vector field X and Y in M , then (M , g) is called an almost contact metric manifold and g is called a compatible metric [14]. In view of (1.2) and (1.3), we get g(ξ, Y ) = η(Y ) (1.4) and g(X, φY ) = −g(φX, Y ). (1.5) An almost contact metric manifold (M n , g) is called a contact metric manifold if dη(X, Y ) = g(X, φY ), (1.6) n for all vector fields X, Y on (M , g). Further, it is considered to be a Sasakian manifold if (1.7) (∇X φ) (Y ) = g(X, Y )ξ − η(Y )X, n for all vector fields X, Y on (M , g). A Riemannian manifold (M n , g) is called semi-symmetric if its curvature tensor R satisfies R(X, Y ).R = 0, (1.8) for all vector fields X, Y on (M n , g), where R(X, Y ) acts on R as a derivation. The studies of semi-symmetric manifolds were done by many authors, (see, [17,19] ). In 2012, Mantica and Molinari [9,10] defined a generalized (0, 2) symmetric Z tensor expressed as Z(X, Y ) = S(X, Y ) + ϕg(X, Y ),
(1.9)
n
where S denotes the Ricci tensor of (M , g) and ϕ is an arbitrary scalar function. It is referred to the generalized Z tensor simply as the Z tensor. In 2013, Mantica and S
Data Loading...
