zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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. (English) Zbl 0782.53025
Classically one has results of L. P. Eisenhart [Trans. Am. Math. Soc. 25, 297-306 (1923)] that a Riemannian manifold admitting a second order symmetric parallel tensor other than a constant multiple of the metric is reducible and H. Levy [Ann. Math. 27 (1926)] that a second order symmetric parallel tensor on a space of constant curvature is a constant multiple of the metric. In [Int. J. Math. Math. Sci. 12, No. 4, 787-790 (1989; Zbl 0696.53012)] the author considered parallel tensor fields that are not necessarily symmetric for both Riemannian and Kähler manifolds. In the present paper the author studies parallel tensor fields on contact metric manifolds. The results of the paper are the following: Ona K-contact manifold a second order symmetric parallel tensor is a constant multiple of the metric. On a Sasakian manifold there are no non-zero parallel 2-forms; this generalizes the result for compact Sasakian manifolds of S. I. Goldberg and the reviewer [J. Differ. Geom. 1, 347-354 (1967; Zbl 0163.439)]. As an application of the first result the author proves that an affine Killing vector field on a compact K-contact manifold without boundary is Killing.

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