10 research outputs found
On compact holomorphically pseudosymmetric K\"ahlerian manifolds
For compact K\"ahlerian manifolds, the holomorphic pseudosymmetry reduces to
the local symmetry if additionally the scalar curvature is constant and the
structure function is non-negative. Similarly, the holomorphic
Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional
assumptions. We construct examples of non-compact essentially holomorphically
pseudosymmetric K\"ahlerian manifolds. These examples show that the compactness
assumption cannot be omitted in the above stated theorem.
Recently, the first examples of compact, simply connected essentially
holomorphically pseudosymmetric K\"ahlerian manifolds are discovered by W.
Jelonek. In his examples, the structure functions change their signs on the
manifold
D'atri spaces of type k and related classes of geometries concerning jacobi operators
In this article we continue the study of the geometry of -D'Atri spaces,
( denotes the dimension of the manifold) began by
the second author. It is known that -D'Atri spaces, are related
to properties of Jacobi operators along geodesics, since she has shown
that , are invariant
under the geodesic flow for any unit tangent vector . Here, assuming that
the Riemannian manifold is a D'Atri space, we prove in our main result that
is also invariant under the geodesic flow if . In addition, other properties of Jacobi operators related to the
Ledger conditions are obtained and they are used to give applications to
Iwasawa type spaces. In the class of D'Atri spaces of Iwasawa type, we show two
different characterizations of the symmetric spaces of noncompact type: they
are exactly the -spaces and on the other hand they are -D'Atri
spaces for some In the last case, they are -D'Atri for all
as well. In particular, Damek-Ricci spaces that are -D'Atri
for some are symmetric.
Finally, we characterize -D'Atri spaces for all as the -spaces (geodesic symmetries preserve the principal curvatures of
small geodesic spheres). Moreover, applying this result in the case of 4%
-dimensional homogeneous spaces we prove that the properties of being a D'Atri
(1-D'Atri) space, or a 3-D'Atri space, are equivalent to the property of being
a -D'Atri space for all .Comment: 19 pages. This paper substitute the previous one where one Theorem
has been deleted and one section has been adde