zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Second order parallel tensors on contact manifolds. II. (English) Zbl 0784.53023
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 ξ denote the characteristic vector field of the contact structure. If the ξ-sectional curvature, K(ξ,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 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 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.

MSC:
53C15Differential geometric structures on manifolds
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)