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Zbl 0784.53023
Sharma, Ramesh
Second order parallel tensors on contact manifolds. II. (English)
[J] C. R. Math. Acad. Sci., Soc. R. Can. 13, No.6, 259-264 (1991). ISSN 0706-1994

In [Algebras Groups Geom. 7, 145-152 (1990; Zbl 0782.53025)] the author proved the following: I) On a $K$-contact manifold a second order symmetric parallel tensor field is a constant multiple of the metric. II) On a Sasakian manifold there are no nonzero parallel 2-forms. In the present paper the author proves a theorem which contains each of the above as a special case. Let $M$ be a contact metric manifold and let $\xi$ denote the characteristic vector field of the contact structure. If the $\xi$-sectional curvature, $K(\xi,X)$, is nowhere vanishing and independent of the direction of $X$, then a second order parallel tensor field on $M$ is a constant multiple of the metric tensor. Examples of non-$K$-contact, contact metric manifolds satisfying the condition may be found in the reviewer's paper with {\it H. Chen} [Bull. Inst. Math., Acad. Sin. 20, No. 4, 379-383 (1992; Zbl 0767.53023)]. In addition to I) and II) being consequences of this result, one also has the following theorem of {\it S. Tanno} [Proc. Japan Acad., Ser. A 43, 581-583 (1967; Zbl 0155.498)] as a corollary: If the Ricci tensor field is parallel on a $K$-contact manifold, then it is an Einstein space.
[D.E.Blair (East Lansing)]

MSC 2000:: *53C15 Geometric structures on manifolds
53C25 Special Riemannian manifolds

Keywords: $K$-contact manifold; $\xi$-sectional curvature; parallel tensor field; Sasakian manifold; parallel 2-forms

Citations: Zbl 0782.53025; Zbl 0767.53023; Zbl 0155.498

Cited in: Zbl 1179.53042

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