On the existence of generalized quasi-Einstein manifolds. (English) Zbl 1249.53063

De, Uday Chand; Mallick, Sahanous

Arch. Math., Brno 47, No. 4, 279-291 (2011).

A non-flat Riemannian manifold \( (M^{n},g) \), \( n> 2 \), is called generalized quasi-Einstein manifold if its Ricci tensor \( S \) is non-zero and satisfies the condition \( S(X,Y) = a\, g(X,Y) + b\, A(X)\, A(Y) + d\, B(X)\, B(Y),\) where \(a,b,d \) are non-zero scalars and \(A, B \) are two non-zero 1-forms such that \( g(A,B)= 0 \), \( \| A\| = \| B\| = 0 \). If \(d=0 \), then the manifold is a quasi-Einstein manifold. In the paper the existence of generalized quasi-Einstein manifold, which is not quasi-Einstein, is proved by describing a non-trivial example in dimension 3.

Reviewer: Josef Janyška (Brno)

MSC:

53C25

Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

quasi-Einstein manifold; generalized quasi-Einstein manifold; manifold of generalized quasi-constant curvature