Cotton tensor, Bach tensor and Kenmotsu manifolds
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Cotton tensor, Bach tensor and Kenmotsu manifolds Amalendu Ghosh1 Received: 29 August 2019 / Accepted: 26 April 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract It is known that Einstein metrics are Bach flat. So, the question may arises whether there exists any Riemannian metric which is Bach flat but not Einstein. In this paper, we answer this by constructing several example on certain class of warped product manifold. Indeed the warped product spaces R × f CPn and R × f CHn with the warping function f (t) = ket (where k is a non zero constant) are non Einstein Bach flat manifolds. These spaces are commonly known as Kenmotsu manifold. Moreover, we prove that a Kenmotsu manifold with parallel Cotton tensor is η-Einstein and Bach flat. Next, we establish that any η-Einstein Kenmotsu manifold of dimension > 3 is Bach flat. Keywords Kenmotsu manifold · Bach tensor · Purely transversal Bach tensor · Warped product · Cotton tensor Mathematics Subject Classification 53C25 · 53C20 · 53C15
1 Introduction In [1], Bach introduced a new type of conformally invariant tensor to study conformal relativity. Consider the following functional on the squared norm of the Weyl conformal curvature W (g) = |W (g)|2 dvg , M4
over a 4-dimensional compact Riemannian manifold M. Then the corresponding critical point of this functional is characterized by the vanishing of a symmetric trace free (0, 2) type tensor Bg , known as Bach tensor. On any Riemannian manifold (M m , g) (m ≥ 4) this is defined by Bg (X , Y ) =
m 1 ((∇ei ∇e j W )(X , ei , e j , Y ) m−3 i, j=1
B 1
Amalendu Ghosh [email protected] Department Of Mathematics, Chandernagore College, Chandannagar, West Bengal 712 136, India
123
A. Ghosh
+
m 1 S(ei , e j )W (X , ei , e j , Y ), m−2
(1.1)
i, j=1
where (ei ), i = 1, . . . , m, is a local orthonormal frame on (M, g), S is the Ricci tensor of type (0, 2) and W is the Weyl tensor of type (0, 4). The (1, 3) type Weyl conformal tensor is defined by 1 {S(Y , Z )X − S(X , Z )Y + g(Y , Z )Q X m−2 r {g(Y , Z )X − g(X , Z )Y }, − g(X , Z )QY } + (m − 1)(m − 2)
W (X , Y )Z = R(X , Y )Z −
(1.2)
where X , Y , Z are arbitrary vector fields on M and R is the curvature tensor, Q is the Ricci operator defined by S(X , Y ) = g(Q X , Y ) and r is the scalar curvature of g. A Riemannian manifold M is said to be Einstein if the Ricci tensor S is a constant multiple of the metric tensor g. Now, from (1.2) and the contraction of Bianchi second identity it follows that m−3 divW = m−2 C, where C is the (0, 3)-type Cotton tensor defined by (see Kuhnel and Rademacher [2]) C(X , Y )Z = (∇ X S)(Y , Z ) − (∇Y S)(X , Z ) 1 [(Xr )g(Y , Z ) − (Y r )g(X , Z )]. − 2(m − 1)
(1.3)
A Riemannian manifold (M m , g) is said to be conformally flat if its conformally related Riemannian metric is locally Euclidean. For m > 3 this is equivalent to W = 0 and for m = 3 this is equivalent to C = 0. Note that the vanishing of the conformal tensor also covers divW = 0 (i.e., C = 0). Th
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