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)
On weakly symmetric Riemannian manifolds. (English) Zbl 1136.53019
A non-flat Riemannian manifold $(M^n,g)$ $(n>2)$ is called weakly symmetric -- and the denotes by $(WS)_n$ -- if its curvature tensor $R$ of type $(0,4)$ satisfies the condition: $$ \aligned & (\nabla_XR)(Y,Z,U,V)=A(X)\cdot R(Y,Z,U,V)+B(Y)\cdot R(X,Z,U,V)\\ &\qquad +C(Z)\cdot R(Y,X,U,V) + D(U)\cdot R(Y,Z,X,V)+E(V)\cdot R(Y,Z,U,X) \endaligned $$ for all vector fields $X,Y,Z,U,V\in \chi (M^n)$, where $A$, $B$, $C$, $D$ and $E$ are 1-forms (non-zero simultaneously) and $\nabla$ is the operator of covariant differentiation with respect to $g$. The present note on $(WS)_n$ consists of 4 sections starting with “Introduction” and “Fundamental results of a $(WS)_n$ $(n>2)$”. In Section 3 on “Conformally flat $(WS)_n$” “ the authors show -- among others -- that a conformally flat $(WS)_n $ $(n>3)$ of non-zero scalar curvature is of hyper quasi-constant curvature (which generalizes the notion of quasi-constant curvature) and also such a manifold is a quasi-Einstein manifold (Theorems 6--9). Finally (Section 4), several examples of $(WS)_n$ of both zero and non-zero scalar curvature are obtained, in particular a manifold $(WS)_n$ $(n\ge 4)$ which is neither locally symmetric nor recurrent, the scalar curvature of which is vanishing (Theorem 12) or non-vanishing and non-constant (Theorem 14), respectively.

53B35Hermitian and Kählerian structures (local differential geometry)
53B05Linear and affine connections